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LIST OF ILLUSTRATIONS XV 


FIG. PAGE 
381, 2. A similar comparison of Diodon and Orthagoriscus  . 
383. The same of various crocodiles: C. porosus, C. americanus Pond 


Notosuchus terrestris . : : : : 2 os 
384. The pelvic girdles of Stegosaurus aad ehiipreninus 754 
385, 6. The shoulder-girdles of Cryptocleidus and of jist Gace ‘ 55 


7 
387. The skulls of Dimorphodon and of Pteranodon ‘ 756 
388-92. The pelves of Archaeopteryx and of Apatornis opnnpenea fad a 
method illustrated whereby intermediate configurations may be 


found by interpolation (G. Heilmann) 5 . 157-9 
393. The same pelves, together with three of the inte maedines or infor 

polated forms  . é ow 
394, 5. Comparison of the ails of we ete Thinoecroree: Op yrachyus 

and Aceratheriwm (Osborn) : : : aol 
396. Occipital views of various extinct MiGnocerosay (ie) : b OS 
397-400. Comparison with each other, and with the skull of Hurachyus, of 

the skulls of Tvtanotheriwm, tapir, horse and rabbit : . 763, 4 


401, 2. Coordinate diagrams of the skulls of Hohippus and of Hquus, with 
various actual and hypothetical intermediate types (Heilmann) . 765-7 


403. A comparison of various human scapulae (Dwight) ; : 09 
404, A human skull, inscribed in Cartesian coordinates ‘ ; sO 
405. The same coordinates on a new Pay adapted to the skull of 

the chimpanzee . : : ee 
- 406. Chimpanzee’s skull, inscribed in ie poirot ce Fig. 405 : tale 


407, 8. Corresponding diagrams of a baboon’s skull. and of a dog’s  . 771,3 


“Cum formarum naturalium et corporalium esse non consistat nisi in 
unione ad materiam, ejusdem agentis esse videtur eas producere cujus est 
materiam transmutare. Secundo, quia cum hujusmodi formae non excedant 
virtutem et ordinem et facultatem principiorum agentium in natura, nulla 
videtur necessitas eorum originem in principia reducere altiora.” Aquinas, 
De Pot. Q. iii, a, 11. (Quoted in Brit. Assoc. Address, Section D, 1911.) 


“..1 would that all other natural phenomena might similarly be deduced 
from mechanical principles. For many things move me to suspect that 
everything depends upon certain forces, in virtue of which the particles of 
bodies, through forces not yet understood, are either impelled together so as 
to cohere in regular figures, or are repelled and recede from one another.” 
Newton, in Preface to the Principia. (Quoted by Mr W. Spottiswoode, 
Brit. Assoc. Presidential Address, 1878.) 


‘““When Science shall have subjected all natural phenomena to the laws of 
Theoretical Mechanics, when she shall be able to predict the result of every 
combination as unerringly as Hamilton predicted conical refraction, or Adams 
revealed to us the existence of Neptune,—that we cannot say. That day 
may never come, and it is certainly far in the dim future. We may not 
anticipate it, we may not even call it possible. But none the less are we 
bound to look to that day, and to labour for it as the crowning triumph of 
Science :—when Theoretical Mechanics shall be recognised as the key to every 
physical enigma, the chart for every traveller through the dark Infinite of 
Nature.” J. H. Jellett, in Brit. Assoc. Address, Section A, 1874. 


ON ss ie 
GROWTH AND FORM 


BY 


DARCY. WENTWORTH EFHOMPSON 


Cambridge : 


at the University Press 


og LF 


“The reasonings about the wonderful and intricate operations 
of nature are so full of uncertainty, that, as the Wise-man truly 
observes, hardly do we guess aright at the things that are wpon 
earth, and with labour do we find the things that are before us.” 
Stephen Hales, Vegetable Staticks (1727), p. 318, 1738. 


PREFATORY NOTE 


HIS book of mine has little need of preface, for indeed it is 

‘all preface’ from beginning to end. I have written it as 

an easy introduction to the study of organic Form, by methods 

which are the common-places of physical science, which are by 

no means novel in their application to natural history, but which 
nevertheless naturalists are little accustomed to employ. 

It is not the biologist with an inkling of mathematics, but 
the skilled and learned mathematician who must ultimately 
deal with such problems as are merely sketched and adumbrated 
here. I pretend to no mathematical skill, but I have made what 
use I could of what tools I had; I have dealt with simple cases, 
and the mathematical methods which I have introduced are of 
the easiest and simplest kind. Elementary as they are, my book 
has not been written without the help—the indispensable help— 
of many friends. Like Mr Pope translating Homer, when I felt 
myself deficient I sought assistance! And the experience which 
Johnson attributed to Pope has been mine also, that men of 
learning did not refuse to help me. 

My debts are many, and I will not try to proclaim them all: 
but I beg to record my particular obligations to Professor Claxton 
Fidler, Sir George Greenhill, Sir Joseph Larmor, and Professor 
A. McKenzie; to a much younger but very helpful friend, 
Mr John Marshall, Scholar of Trinity; lastly, and (if I may say 
- so) most of all, to my colleague Professor William Peddie, whose 
advice has made many useful additions to my book and whose 
criticism has spared me many a fault and blunder. 

I am under obligations also to the authors and publishers of 
many books from which illustrations have been borrowed, and 
especially to the following :— 

To the Controller of H.M. Stationery Office, for leave to 
reproduce a number of figures, chiefly of Foraminifera and of 
Radiolaria, from the Reports of the Challenger Expedition. 


v1 PREFATORY NOTE 


To the Council of the Royal Society of Edinburgh, and to that 
of the Zoological Society of London :—the former for letting me 
reprint from their Transactions the greater part of the text and 
illustrations of my concluding chapter, the latter for the use of a 
number of figures for my chapter on Horns. 

To Professor EK. B. Wilson, for his well-known and all but 
indispensable figures of the cell (figs. 42—51, 53); to M. A. Prenant, 
for other figures (41, 48) in the same chapter; to Sir Donald 
MacAlister and Mr Edwin Arnold for certain figures (335—7), 
and to Sir Edward Schafer and Messrs Longmans for another (334), 
illustrating the minute trabecular structure of bone. To Mr 
Gerhard Heilmann, of Copenhagen, for his beautiful diagrams 
(figs. 388-93, 401, 402) included in my last chapter. To Pro- 
fessor Claxton Fidler and to Messrs Griffin, for letting me use, 
with more or less modification or simplification, a number of 
illustrations (figs. 339—346) from Professor Fidler’s Textbook of 
Bridge Construction. To Messrs Blackwood and Sons, for several 
cuts (figs. 127-9, 131, 173) from Professor Alleyne Nicholson’s 
Palaeontology; to Mr Heinemann, for certain figures (57, 122, 123, 
205) from Dr Stéphane Leduc’s Mechanism of Infe; to Mr A. M. 
Worthington and to Messrs Longmans, for figures (71, 75) from 
A Study of Splashes, and to Mr C. R. Darling and to Messrs EH. 
and 8. Spon for those (fig. 85) from Mr Darling’s Liquid Drops 
and Globules. To Messrs Macmillan and Co. for two figures 
(304, 305) from Zittel’s Palaeontology; to the Oxford University 
Press for a diagram (fig. 28) from Mr J. W. Jenkinson’s Hxperi- 
_mental Embryology; and to the Cambridge University Press for 
a number of figures from Professor Henry Woods’s Invertebrate 
Palaeontology, for orte (fig. 210) from Dr Willey’s Zoological Results, 
and for another (fig. 321) from “ Thomson and Tait.” 

Many more, and by much the greater part of my diagrams, 
I owe to the untiring help of Dr Doris L. Mackinnon, D.Sc., and 
of Miss Helen Ogilvie, M.A., B.Sc., of this College. 


D’ARCY WENTWORTH THOMPSON. 


UNIVERSITY COLLEGE, DUNDEE. 
December, 1916. 


XVII. 


CONTENTS 


INTRODUCTORY Ss : ‘: 5 A 3 ° 3 A 
On MAGNITUDE . ‘ ‘ 5 ‘ ¢ A : 
THe Rate or GRowtH ‘ 3 : 4 A A 


On THE INTERNAL FoRM AND STRUCTURE OF THE CELL 
THE Forms oF CELLS 

A Notre on ADSORPTION 

THe Forms or TISSUES, OR CELL-AGGREGATES 

THE SAME (continued) . A ; : j agate 

On CONCRETIONS, SPICULES, AND SPICULAR SKELETONS 
A PARENTHETIC NOTE ON GEODETICS 

THE LOGARITHMIC SPIRAL 

THE SPIRAL SHELLS OF THE FORAMINIFERA 


THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH 
A NoTE on TORSION 


On LEAF-ARRANGEMENT, OR PHYLLOTAXIS 


ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW 
STRUCTURES . ; 5 5 


On Form AND MECHANICAL EFFICIENCY . 


On THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON 
oF RELATED FoRMS 


EPILOGuE 


INDEX 


og iw 


PAGE 


LIST OF ILLUSTRATIONS 


FIG. PAGE 
1. Nerve-cells, from larger and smaller animals (Minot, after Irving 

Hardesty) . - : “ 37 

2. Relative magnitudes of some nantes organisms ‘(Gearon : ‘ 39 

3. Curves of growth in man (Quetelet and Bowditch) : - : 61 


4, 5. Mean eas increments of stature and weight in man (do.) . 66, 69 
6. The ratio, throughout life. of female weight to male (do.) . - 71 
7-9. Curves of growth of child, before and after birth (His and Riissow) 74-6 


10. Curve of growth of bamboo (Ostwald. after Kraus) : 2 urs 
11. Coefficients cf variability in human stature (Boas and Wiesler) : 80 
12. Growth in weight cf mouse (Wolfgang Ostwald) . : 4 2 83 
13. Dea, of silkwerm (Luciani and Lo Monaco) : : 4 , : 84 
14. Do. cf tadpole (Ostwald, after Schaper) . : 3 85 
15. Larval eels, or Leptocephali, arid young elver (Joh. Schmidt) ee TSO 
16. Growth in length of Spirogyra (Hofmeister) . : ; ; 2 87 
17. Pulsations of growth in Crecus (Bose)  . : 88 
18. Relative growth of brain, heart and body of man ( Gidetelct) : 90 
19. Ratio of stature to span of arms (do.) . : : 5 : 94 
20. Rates of growth near the tip of a bean-root (Sachs) F : : 96 
21, 22. The weight-length ratio of the plaice, and its annual periodic 

changes : ; : : § : 99, 100 
23. Variability of tail- pee in earwigs (eatoan) ‘ ; : => Oz: 
24. Variability of body-length in plaice - F eo EOE 
25. Rate of growth in plants in relation to iamrpcestne (Sachs) i? GS 
26. Do. in maize, observed (K6ppen), and calculated curves . : ~ tee 
27. Do. in roots of peas (Miss I. Leitch) : : 113 
28, 29. Rate of growth of frog in relation to ieueetonc (iekicaaree 

after O. Hertwig), and calculated curves of do. : : - 115,6 
30. Seasonal fluctuation of rate of growth in man (Dafiner) P EEG 
31. Do. in the rate of growth of trees (C. E. Hall) . : : - 220 
32. Long-period fluctuation in the rate of growth of Arizona trees 

(A. E. Douglass) : : : : : ee oe 
33, 34. The varying form of oe pneeiee NAveonial: in relation to 

salinity (Abonyi) . : . 128.9 


35-39. Curves of regenerative eens in Gaiseles ‘ails (M. L. Durbis) 140-145 
40. Relation between amount of tail removed, amount restored, and 


time required for restoration (M. M. Ellis) ; . 148 
41. Caryokinesis in trout’s egg (Prenant, after Prof. P. Bouin) : 7 al G9 
42-51. Diagrams of mitotic cell-division (Prof. E. B. Wilson). . 171-5 


52. Chromosomes in course of splitting and separation (Hatschek and 
Flemming) . : : ‘ : - - : : : . , 180 


LIST OF ILLUSTRATIONS 1x 


FIG. PAGE 
53. Annular chromosomes of mole-cricket (Wilson, after vom Rath) . 181 
54-56. Diagrams illustrating a hypothetic field of force in caryokinesis 

(Prof. W. Peddie) : i : & F . 182-4 
57. An artificial figure of enor fede} ‘ : , i, 86 
58. A segmented egg of Cerebratulus (Prenant, after Coe) ; : =» 189 
59. Diagram of a field of force with two like poles. : : . 189 
60. A budding yeast-cell ASS AEWA en eH Fale a Wieeek tie, Ce 
61. The roulettes of the conic sections . ; Billie 
62. Mode of development of an unduloid from a amuncal fae rl) 

63-65. Cylindrical, unduloid, nodoid and catenoid oil-globules (Plateau) 222, 3 
66. Diagram of the nodoid, or elastic curve , 224 
67. Diagram of a cylinder capped by the corresponding paaiea of a ieee 226 
68. <A liquid cylinder breaking up into spheres’. : wee 
69. The same phenomenon in a protoplasmic cell of ney : . 234 
70. Some phases of a splash (A. M. Worthington) ; : : . 235 
71. <A breaking wave (do.) : : : : : ; 3, 280 
72. The calycles of some campanularian soophaies : : ; Se pee-5 
73. A flagellate monad, Distigma proteus (Saville Kent) ‘ : . 246 
74. Noctiluca miliaris, diagrammatic : : : : . 246 
75. Various species of Vorticella (Saville Kent ad, bther\ ; ; » 247 
76. Various species of Salpingoeca (do.) : ; ; . 248 
77. Species of Tintinnus, Dinobryon and Codonelia an ; : 5 248 

78. The tube or cup of Vaginicola BM) irae : ; : F . 248 
79. The same of Folliculina . : : : : : : eae 280 
80. Trachelophyllum (Wreszniowski) ; : ‘ : ; : . 249 
81. Trichodina pediculus ; : : F : 3 ; Bay? 
82. Dinenympha gracilis (Leidy) . : : : ; : . . 2a 
83. A “collar-cell” of Codosiga : : E : : : . » 254 
84. Various species of Lagena (Brady) ©. e200 
85. Hanging drops, to illustrate the umaatenel fora (C. 'R. Darling) ci Peta 
86. Diagram of a fluted cylinder . : A : : d . 260 
87. Nodosaria scalaris (Brady) : : 5 PAOPS 
88. Fluted and pleated gonangia of pete Chee onl ees (Allman) a 262 
89. Various species of Nodosaria, Sagrina and Rheophax (Brady) . . 263 
90. Trypanosoma tineae and Spirochaeta anodontae, to shew undulating 

membranes (Minchin and Fantham) ‘ : : 2 266 

91. Some species of Trichomastix and Trichomonas (Kofoid) 5 40 LOI 

92. Herpetomonas assuming the undulatory membrane of a Trypanosome 
(D. L. Mackinnon) . ‘ ; 5 3 ; : . 268 
93. Diagram of a human blood- poruncle : 271 

94. Sperm-cells of decapod crustacea, Inachus and Calathied (Koltzofi) 273 
95. The same, in saline solutions of varying density (do.) . 5 . 274 
96. A sperm-cell of Dromia (do.) . : 275 


97. Chondriosomes in cells of kidney and pancreas (Barat one Mathews) 285 
98. Adsorptive concentration of eae aes salts in various plant-cells 


(Macallum) : : : - 290 
99-101. Equilibrium of ae Jeon in a eniiag or : . 294, 5 
102. Plateau’s “‘bourrelet” in plant-cells; diagrammatic (Berthold) . 298 


103. Parenchyma of maize, shewing the same phenomenon . . »298 


x LIST OF ILLUSTRATIONS 


FIG. 


PAGE 


104, 5. Diagrams of the partition-wall between two soap-bubbles . 299, 300 


106. Diagram of a partition in a conical cell : 300 
107. Chains of cells in Nostoc, Anabaena and other low aleae 300 
108. Diagram of a symmetrically divided soap-bubble - aol 
109. Arrangement of partitions in dividing spores of Pellia (Camppell. 302 
110. Cells of Dictyota (Reinke) 303 
111, 2. Terminal and other cells of Chari, Rade reane ntheciditird of te 303 
113. Diagram of cell-walls and partitions under various conditions of 
tension 304 
114, 5. The partition- \ saettned of fee dorconmiented Bubbles 307,8 
116. Diagram of four interconnected cells or bubbles 309 
117. Various configurations of four cells in a frog’s egg (Rauber) . 311 
118. Another diagram of two conjoined soap-bubbles ‘ 313 
119. A froth of bubbles, shewing its outer or “epidermal” layer . 314 
120. A tetrahedron, or tetrahedral system, shewing its centre of symmetry 317 
121. A group of bexagonal cells (Bonanni) 319 
122, 3. Artificial cellular tissues (Leduc) . 320 
124. Epidermis of Girardia (Goebel) 321 
125. Soap-froth, and the same under compression (Bhasnblen) 322 
126. Epidermal cells of Elodea canadensis (Berthold) 322 
127. Lithostrotion Martini (Nicholson) : 325 
128. Cyathophyllum hexagonum (Nicholson, ter Zittel) : 325 
129. Arachnophyllum pentagonum (Nicholson) . 326 
130. Heliolites (Woods) ; : . 326 
131. Confluent septa in hanna aen al Gomosmis (Nicholson, after 
Zittel) 327 
132. Geometrical construction 3 a hes? s cell : 330 
133. Stellate cells in the pith of a rush; diagrammatic 335 
134. Diagram of soap-films formed in a cubical wire skeleton (Plateau) 337 
135. Polar furrows in systems of four soap-bubbles (Robert) 341 
136-8. _ Diagrams illustrating the division of a cube by partitions of minimal 
area. : : : - 347-50 
139. Cells from hairs of Sihiabeloria (Berthold) 351 
140. The bisection of an isosceles triangle by minimal patties Hit ene S52: 
141. The similar partitioning of spheroidal and conical cells . 353 
142. S-shaped partitions from cells of algae and mosses (Reinke and others) 355 
143. Diagrammatic explanation of the S-shaped partitions 356 
144. Development of Hrythrotrichia (Berthold) 359 
145. Periclinal, anticlinal and radial partitioning of a quadieee 359 
146. Construction for the minimal partitioning of a quadrant 361 
147. Another diagram of anticlinal and periclinal partitions . 362 
148. Mode of segmentation of an artificially flattened frog’s egg 
(Roux) : 363 
149. The bisection, by eee ears aE a prism a al snale : 364 
150. Comparative diagram of the various modes of bisection of a prismatic 
sector 365 
151. Diagram of the further sows ge ete) two hale of a quareeael cell» 367 
152. Diagram of the origin of an epidermic layer of cells 370 
153. <A discoidal cell dividing into octants 371 


LIST OF ILLUSTRATIONS xl 


FIG. PAGE 
154. A germinating spore of Riccia (after Campbell), to shew the manner 

of space-partitioning in the cellular tissue ‘ 
155, 6. Theoretical arrangement of successive partitions in a aR ioidsl cell 373 


157. Sections of a moss-embryo (Kienitz-Gerloff) . : 374 

158. Various possible arrangements of De ae in groups of fone toe a 
cellsie. : 5; : .+) 315 
159. Three modes of pecans in a eat of six alls ; eke 
160, 1. Segmenting eggs of T'rochus (Robert), and of Cynthia (C ‘onlin F eone 
162. Section of the apical cone of Salvinia (Pringsheim) ; 377 
163,4. Segmenting eggs of Pyrosoma (Korotneff), and of Echinus @omese h) 377 
165. Segmenting egg of a cephalopod (Watase) ; ‘ a Bilis: 

166, 7. Eggs segmenting under pressure: of Echinus and Wer els (Driesch), 
and of a frog (Roux) ; 378 

168. Various arrangements of a group of Soles € ei on he pratace of a ae s 
egg (Rauber) ; : 381 

169. Diagram of the partitions and interacial contae és ina neveent of e ipth 
cells: : ; 6 RG 1-33) 
170. Various modes of sess bites of cane vil neve (Roux) : . 9384 
171. Forms, or species, of Asterolampra (Greville) . : 2 ; .. 386 
172. Diagrammatic section of an alcyonarian polype  . . : = peed 
173, 4. Sections of Heterophyllia (Nicholson and Martin Duncan) . 388, 9 
175. Diagrammatic section of a ctenophore (Hucharis) . : < . 391 
176, 7. Diagrams of the construction of a Pluteus larva : . 392,3 

178, 9. Diagrams of the eka of stomata, in Sedum and in the 
hyacinth. : : : . 394 
180. Various spores and pallens orains (Berthold al deliers) : . = 1396 
181. Spore of Anthoceros (Campbell) ‘ : : 5 a)T 

182, 4,9. Diagrammatic modes of division of a cell Gude perii conditions 
of asymmetry . : A : . 400-5 
183. Development of the embryo ae Shag um we arrpbell) ; j . 402 
185. The gemma of a moss (do.) . ; : , : : ; . 403 
186. The antheridium of Riccia (do.) 3 5 3 ; . 404 
187. Section of growing shoot of Selaginella, Teenie: : : . 404 
188. An embryo of Jungermannia (Kienitz-Gerloff) : : u . 404 
190. Development of the sporangium of Osmunda (Bower). 5 . 406 
191. Embryos of Phascum and of Adiantum (Kienitz-Gerlot) ; . 408 
192. A section of Girardia (Goebel) ‘ : : ; : 5 . 408 
193. An antheridium of Pteris (Strasburger) . ; ; eae 
194. Spicules of Siphonogorgia and Anthogorgia (Studer) ; E 413 
195-7. Calcospherites, deposited in white of egg (Harting) . : _ 421,2 
198. Sections of the shell of Mya (Carpenter) F : = 22 
199. Concretions, or spicules, artificially deposited in ceelage (Harting) 423 
200. Further illustrations of aleyonarian spicules: Hunicea (Studer) . 424 
201-3. Associated, aggregated and composite calcospherites (Harting) . 425,6 
204. Harting’s “conostats” : 2 : : : : - : “eed 
205. Liesegang’s rings (Leduc) : ; ; : : . 428 
206. Relay-crystals of common salt een) : : : ; 4429 
207. Wheel-like crystals in a colloid medium (do.) : - 429 


208. A concentrically striated calcospherite or spherocrystal (Hantittg) . 432 


xii LIST OF ILLUSTRATIONS 


FIG. 


PAGE 
209. Otoliths of plaice, shewing “‘age-rings” (Wallace) . : : . 432 
210. Spicules, or calcospherites, of Astrosclera (Lister) . : . 436 
211, 2. C- and S-shaped spicules of sponges and holothurians (olla and 

Théel) , : ; ; : ; : ; ; . 442 
215. An amphidisc of al Gnens : 442 


214-7. Spicules of calcareous, ioirechinellid and horace cli eae aad 
of various holothurians (Haeckel, Schultze, Sollas and Théel) 445-452 
218. Diagram of a solid body confined by surface-energy to a liquid 


boundary-film . : ; : : ; : . 460 
219. Astrorhiza limicola and arenaria eee : ; ; ecu aes 
220. A nuclear “reticulum plasmatique” (Carnoy) . 3 2 ; . 468 
221. A sphervical radiolarian, Aulonia hexagona (Haeckel) a : . . 469 
222. Actinomma arcadophorum (do.) ; : - 2 c - - 469 
223. Hthmosphaera conosiphonia (do.) : : ‘ . 470 
224, Portions of shells of Cenosphaera favosa and vesparia (do.) : =“) 0 
225. Aulastrum triceros (do.)  . : : : a ; . A471 
226. Part of the skeleton of Cannes (do.) t 5 : : . 472 
227. <A Nassellarian skeleton, Callimitra carolotae (do.) . ; ; .- 472 
228, 9. Portions of Dictyocha stapedia (do.)  . ; ; : . 474 
230. Diagram to illustrate the conformation of Cilngtiva c : a IG 
231. Skeletons of various radiolarians (Haeckel)  . ; 5 « AT9 
232. Diagrammatic structure of the skeleton of Dorataspis (do.) : . 481 
233, 4. Phatnaspis cristata (Haeckel), and a diagram of the same . . 483 
235. Phractaspis prototypus (Haeckel) : : ; . 484 
236. Annular and spiral thickenings in the walls of noe cells : . 488 
237. A radiograph of the shell of Nautilus (Green and Gardiner) . . 494 
238. A spiral foraminifer, Globigerina (Brady) : : 495 
239-42. Diagrams to illustrate the development or growth of a eerie. 

spiral . : ; : : : ‘ : : : 407-501 
243. A helicoid and a eee Cymer : : : ; : - - 502 
244, An Archimedean spiral  . : : . 503 
245-7. More diagrams of the fine ete Aa a laeatchmic Epirall: . 505, 6 


248-57. Various diagrams illustrating the mathematical theory of gnomons 508-13 
258. <A shell of Haliotis, to shew how each increment of the shell constitutes 


a gnomon to the preexisting structure. 514 
259, 60. Spiral foraminifera, Pulvinulina and Cristellaria, co fies = ine 

same principle . : 3 : ; : . 514,5 
261. Another diagram of a eevee ea : Seay 
262. A diagram of the logarithmic spiral of Riau (ivigecleys Z eas Og 
263, 4. Opercula of Turbo and of Nerita (Moseley) : ‘ : - 821,.2 
265. <A section of the shell of Melo ethiopicus ; : 525 
266. Shells of Harpa and Doliwm, to illustrate generating curves aad 

generating spirals : , - : : ‘ : ‘ -. 526 
267. D’Orbigny’s Helicometer . é : : . ; ees) 
268. Section of a nautiloid shell, to shew fae ““protoconch”’ . 5 531 
269-73. Diagrams of logarithmic spirals, of various angles. , 532-5 
274, 6, 7. Constructions for determining the angle of a logarithmic spiral 537, 8 
275. An ammonite, to shew its corrugated surface-pattern  . : aor 20.0 


278-80. Illustrations of the “angle of retardation” : ¥ 4 . 542-4 


LIST OF ILLUSTRATIONS 


FIG. 

281. A shell’of Macrescaphites, to shew change of curvature 

282. Construction for determining the length of the coiled spire 

283. Section of the shell of Triton corrugatus (Woodward) 

284. Lamellaria perspicua and Sigaretus haliotoides (do.) 

285, 6. Sections of the shells of Terebra maculata and Trochus nilonie Us 

287-9. Diagrams illustrating the lines of growth on a lamellibranch shell | 

290. Caprinella adversa (Woodward) 4 : 

291. Section of the shell of Productus (Woods) 

292. The “skeletal loop” of Terebratula (do.) . : 

293, 4. The spiral arms of Spirifer and of Atrypa (do.) : 

295-7. Shells of Cleodora, Hyalaea and other pteropods (Boas) 

298, 9. Coordinate diagrams of the shell-outline in certain pteropods 

300. Development of the shell of Hyalaea tridentata (Tesch) . 

301. Pteropod shells, of Cleodora and Hyalaea, viewed from the side ( Boas) 

302, 3. Diagrams of septa in a conical shell 

304. <A section of Nautilus, shewing the logarithmic spirals of the eats 40 
which the shell-spiral is the evolute 

305. Cast of the interior of the shell of Nautilus, to ee the eonigine ee 
the septa at their junction with the shell-wall : 

306. Ammonites Sowerbyi, to shew septal outlines (Zittel, after Saini 
and Déderlein) ; 

307. Suture-line of Pinacoceras (Zittel, aiken Hanes) 

308. Shells of Hastigerina, to shew the “mouth” (Brady) 

309. Nummulina antiquior (V. von Moller) : 

310. Cornuspira foliacea and Operculina complanata (Brady) 

311. Miliolina pulchella and linnaeana (Brady) 

312, 3. Cyclammina cancellata (do.), and diagrammatic figure af the, same 

314. Orbulina universa (Brady) 

315. Cristellaria reniformis (do.) 

316. Discorbina bertheloti (do.) 

317. Teatularia trochus and concava gant ‘ 

318. Diagrammatic figure of a ram’s horns (Sir V. Brooke) 

319. Head of an Arabian wild goat (Sclater) . 

320. Head of Ovis Ammon, shewing St Venant’s curves 

321. St Venant’s diagram of a triangular prism under torsion (fiomecn 

and Tait) : 

322. Diagram of the same phienoienensts in a ram’s alae 

323. Antlers of a Swedish elk (Lénnberg) ‘ 

324. Head and antlers of Cervus duvauceli (eydeebenye 

325, 6. Diagrams of spiral phyllotaxis (P. G. Tait) ; 

327. Further diagrams of phyllotaxis, to shew how various spiral papeat: 
ances may arise out of one and the same angular leaf-divergence 

328. Diagrammatic outlines of various sea-urchins . n é 

329, 30. Diagrams of the angle of branching in blood- eect (Hess) 

331, 2. Diagrams illustrating the flexure of a beam 

333. An example of the mode of arrangement of bast-fibres in a late -stem 
(Schwendener) 

334. Section of the head of a Four ta: chew its imapeculas pecheture 
(Schafer, after Robinson) 


XIV LIST OF ILLUSTRATIONS 


FIG. PAGE 
335 Comparative diagrams of a crane-head and the head of a femur ~~ 

(Culmann and H. Meyer) . 3 682 
336. Diagram of stress-lines in the human foot (Sir D. MacAlister dea 

H. Meyer) . - : : : : . 684 
337. Trabecular structure of ine 0s pare (do.) 5 : - 685 
338. Diagram of shearing-stress in a loaded pillar . : : : . 686 
339. Diagrams of tied arch, and bowstring girder (Fidler) . : . 693 
340, 1. Diagrams of a bridge: shewing proposed span, the corresponding 

stress-diagram and reciprocal plan of construction (do.) . 2.696 
342. A loaded bracket and its reciprocal construction-diagram (Culmann). 697 
343, 4. A cantilever bridge, with its reciprocal diagrams (Fidler) . ous 
345. <A two-armed cantilever of the Forth Bridge (do.) . ; coe Oe 
346. A two-armed cantilever with load distributed over two pier- eae 

as in the quadrupedal skeleton : 700 
347-9. Stress-diagrams, or diagrams of bending Toman in fe backbones 

of the horse, of a Dinosaur, and of Tvtanotherium . : . 701-4 
350. The skeleton of Stegosaurus : A ila eeaOd 
351. Bending-moments in a beam Site feed ends, to illustrate the 

mechanics of chevron-bones : oo 8 MU) 
352, 3. Coordinate diagrams of a circle, and its dekdrmiation inte an 

ellipse : 729 
354. Comparison, by means oe Orton eae. of the cannon- bones 

of various ruminant animals. : a 720) 
355, 6. Logarithmic coordinates, and the circle of Fig. 352 yea fee 729, 31 
357, 8. Diagrams of oblique and radial coordinates . : : ») iit 
359. Lanceolate, ovate and cordate leaves, compared by the fap of radial 

coordinates : ; : 5 2 : : : at hee 
360. <A leaf of Begonia daedalea : : F : 4 ‘ of hos 
361. A network of logarithmic spiral ponmeinaeee ‘ : 5 a a 
362, 3. Feet of ox, sheep and giraffe, compared by means of Cartesian 

coordinates : - 738, 40 


364, 6. “ Proportional dineearae: of ON Si enore ( Albert Diirer) 740, 2 
365. Median and lateral toes of a tapir, compared by means of rectangular 


and oblique coordinates . - - 741 
367, 8. A comparison of the copepods Gitians ad ein ; 742 
369. The carapaces of certain crabs, Geryon, Corystes and others, Soe 
by means of rectilinear and curvilinear coordinates : 744 
370. A comparison of certain amphipods, Harpinia, ae oad 
Hyperia : 746 
371 The calycles of aor pineaon sn Sues tiecnibed in corre- 
sponding Cartesian networks  . y 747 
372. The calycles of certain species of Aglaophenia, rade compared By 
means of curvilinear coordinates . 748. 
373, 4. The fishes Argyropelecus and Sternoptyz, Garapared by means of 
rectangular and oblique coordinate systems . 748 
375, 6. Scarus and Pomacanthus, similarly compared by means of reat 
angular and coaxial systems . 749 


377-80. Acomparison of the fishes Polyprion, Pecul op eenanihin Sionipeeae 
and Antigonia . < 5 2 : : : : : 5 7a0 


CHAPTER I 
INTRODUCTORY 


Of the chemistry of his day and generation, Kant declared 
that it was ‘“‘a science, but not science,’—‘“eine Wissenschaft, 
aber nicht Wissenschaft”; for that the criterion of physical 
science lay in its relation to mathematics. And a hundred years 
later Du Bois Reymond, profound student of the many sciences 
on which physiology is based, recalled and reiterated the old 
saying, declaring that chemistry would only reach the rank of 
science, in the high and strict sense, when it should be found 
possible to explain chemical reactions in the light of their causal 
relation to the velocities, tensions and conditions of equilibrium 
of the component molecules; that, in short, the chemistry of the 
future must deal with molecular mechanics, by the methods and 
in the strict language of mathematics, as the astronomy of Newton 
and Laplace dealt with the stars in their courses. We know how 
great a step has been made towards this distant and once hopeless 
goal, as Kant defined it, since van’t Hoff laid the firm foundations 
of a mathematical chemistry, and earned his proud epitaph, 
Physicam chemiae adiunxit*. 

We need not wait for the full realisation of Kant’s desire, in 
order to apply to the natural sciences the principle which he 
urged. Though chemistry fall short of its ultimate goal in mathe- 
matical mechanics, nevertheless physiology is vastly strengthened 
and enlarged by making use of the chemistry, as of the physics, 
of the age. Little by little it draws nearer to our conception of 
a true science, with each branch of physical science which it 

* These sayings of Kant and of Du Bois, and others like to them, have been 


the text of many discourses: see, for instance, Stallo’s Concepts, p. 21, 1882; Hober, 
Biol. Centralbl. x1x, p. 284, 1890, ete. Cf. also Jellett, Rep. Brit. Ass. 1874, p. 1. 


T. G. 1 


2 INTRODUCTORY [CH. 


brings into relation with itself: with every physical law and every 
mathematical theorem which it learns to take into its employ. 
Between the physiology of Haller, fine as it was, and that of 
Helmholtz, Ludwig, Claude Bernard, there was all the difference 
in the world. 

As soon as we adventure on the paths of the physicist, we 
learn to weigh and to measure, to deal with time and space and 
mass and their related concepts, and to find more and more 
our knowledge expressed and our needs satisfied through the 
concept of number, as in the dreams and visions of Plato and 
Pythagoras; for modern chemistry would have gladdened the 
hearts of those great philosophic dreamers. 

But the zoologist or morphologist has been slow, where the 
physiologist has long been eager, to invoke the aid of the physical 
or mathematical sciences; and the reasons for this difference lie 
deep, and in part are rooted in old traditions. The zoologist has 
scarce begun to dream:of defining, in mathematical language, even 
the simpler organic forms. When he finds a simple geometrical 
construction, for instance in the honey-comb, he would fain refer 
it to psychical instinct or design rather than to the operation of 
physical forces; when he sees in snail, or nautilus, or tiny 
foraminiferal or radiolarian shell, a close approach to the perfect 
sphere or spiral, he is prone, of old habit, to beheve that it is 
after all something more than a spiral or a sphere, and that in 
this ““something more” there hes what neither physics nor 
mathematics can explain. In short he is deeply reluctant to 
compare the living with the dead, or to explain by geometry or 
by dynamics the things which have their part in the mystery of 
life. Moreover he is little inclined to feel the need of such 
explanations or of such extension of his field of thought. He is 
not without some justification if he feels that in admiration of 
nature’s handiwork he has an horizon open before his eyes as 
wide as any man requires. He has the help of many fascinating 
theories within the bounds of his own science, whieh, though 
a little lacking in precision, serve the purpose of ordering his 
thoughts and of suggesting new objects of enquiry. His art of 
classification becomes a ceaseless and an endless search after the 
blood-relationships of things living, and the pedigrees of things 


1] THE FINAL CAUSE 3 


dead and gone. The facts of embryology become for him, as 
Wolff, von Baer and Fritz Miiller proclaimed, a record not only 
of the life-history of the individual but of the annals of its race. 
The facts of geographical distribution or even of the migration of 
birds lead on and on to speculations regarding lost continents, 
sunken islands, or bridges across ancient seas. Hvery nesting 
bird, every ant-hill or spider's web displays its psychological 
problems of instinct or intelligence. Above all, in things both 
great and small, the naturalist is rightfully impressed, and finally 
engrossed, by the peculiar beauty which is manifested in apparent 
fitness or ““adaptation,’—the flower for the bee, the berry for the 
bird. 

Time out of mind, it has been by way of the “final cause,” 
by the teleological concept of “end,” of “purpose,” or of “design,” 
in one or another of its many forms (for its moods are many), 
that men have been chiefly wont to explain the phenomena of 
the living world; and it will be so while men have eyes to see 
and ears to hear withal. With Galen, as with Aristotle, it was 
the physician’s way; with John Ray, as with Aristotle, it was the 
naturalist’s way; with Kant, as with Aristotle, it was the philo- 
sopher’s way. It was the old Hebrew way, and has its splendid 
setting in the story that God made “every plant of the field before 
it was in the earth, and every herb of the field before it grew.” 
It is a common way, and a great way; for it brings with it a 
glimpse of a great vision, and it lies deep as the love of nature 
in the hearts of men. 

Half overshadowing the “efficient” or physical cause, the 
argument of the final cause appears in eighteenth century physics, 
in the hands of such men as Euler* and Maupertuis, to whom 
Leibniz} had passed it on. Half overshadowed by the me- 
chanical concept, it runs through Claude Bernard’s Legons sur les 


* “Quum enim mundi universi fabrica sit perfectissima, atque a Creatore 
sapientissimo absoluta, nihil omnino in mundo contingit in quo non maximi 
minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin 
omnes mundi effectus ex causis finalibus, ope methodi maximorum et minimorum, 
aeque feliciter determinari queant atque ex ipsis causis efficientibus.” Methodus 
inveniendi, etc. 1744 (cit. Mach, Science of Mechanics, 1902, p. 455). 

+ Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes finales..., 
eest de 1a qu’il faut tout déduire en Physique.” 

]—2 


hee ere INTRODUCTORY [cH. 


phénomeénes de la Vie*, and abides in much of modern physio- 
logy+. Inherited from Hegel, it dominated Oken’s Naturphilosophie 
and lingered among his later disciples, who were wont to liken 
the course of organic evolution not to the straggling branches of 
a tree, but to the building of a temple, divinely planned, and the 
crowning of it with its polished minarets f. 

It is retained, somewhat crudely, in modern embryology, by 
those who see in the early processes of growth a significance 
“rather prospective than retrospective,’ such that the embryonic 
phenomena must be “referred directly to their usefulness in 
building the body of the future animal§” :—which is no more, and 
no less, than to say, with Aristotle, that the organism is the TéXos, 
or final cause, of its own processes of generation and development. 
It is writ large in that Entelechy|| which Driesch rediscovered, 
and which he made known to many who had neither learned of it 
from Aristotle, nor studied it with Leibniz, nor laughed at it with 
Voltaire. And, though it is in a very curious way, we are told that 
teleology was “‘refounded, reformed or rehabilitated] ” by Darwin's 
theory of natural selection, whereby “‘every variety of form and 
colour was urgently and absolutely called upon to produce its 
title to existence either as an active useful agent, or as a survival” - 
of such active usefulness in the past. But in this last, and very 
important case, we have reached a “teleology” without a téXos, 


* Cf. p. 162. “La force vitale dirige des phénomeénes qu’elle ne produit pas: 
les agents physiques produisent des phénomeénes qu’ils ne dirigent pas.” 

+ It is now and then conceded with reluctance. Thus Enriques, a learned 
and philosophic naturalist, writing “della economia di sostanza nelle osse cave” 
(Arch. f. Entw. Mech. xx, 1906), says ‘‘una certa impronta di teleologismo qua e 1a 
é rimasta, mio malgrado, in questo scritto.” 

t Cf. Cleland, On Terminal Forms of Life, J. Anat. and Phys. xvi, 1884. 

§ Conklin, Embryology of Crepidula, Journ. of Morphol. x1, p. 203, 1897; 
Lillie, F. R., Adaptation in Cleavage, Woods Holl Biol. Lectures, pp. 43-67, 1899. 

|| I am inclined to trace back Driesch’s teaching of Entelechy to no less a 
person than Melanchthon. When Bacon (de Augm. Iv, 3) states with disapproval 
that the soul “has been regarded rather as a function than as a substance,” R. L. 
Ellis points out that he is referring to Melanchthon’s exposition of the Aristotelian 
doctrine. For Melanchthon, whose view of the peripatetic philosophy had long 
great influence in the Protestant Universities, affirmed that, according to the true 
view of Aristotle’s opinion, the soul is not a substance, but an évre\éxeva, or 
function. He defined it as dtvauis quaedam ciens actiones—a description all but 
identical with that of Claude Bernard’s ‘force vitale.” 

{| Ray Lankester, Encycl. Brit. (9th ed.), art. “Zoology,” p. 806, 1888. 


1] OF TELEOLOGY AND MECHANISM i) 


as men like Butler and Janet have been prompt toshew: a teleology 
in which the final cause becomes little more, if anything, than the 
mere expression or resultant of a process of sifting out of the 
good from the bad, or of the better from the worse, in short of 
a process of mechanism*. The apparent manifestations of “ pur- 
pose” or adaptation become part of a mechanical philosophy, 
according to which “chaque chose finit toujours par s’accommoder 
a son milieuf.”’ In short, by a road which resembles but is not 
the same as Maupertuis’s road, we find our way to the very world 
in which we are living, and find that if it be not, it is ever tending 
to become, “the best of all possible worlds.” 

But the use of the teleological principle is but one way, not 
the whole or the only way, by which we may seek to learn how 
things came to be, and to take their places in the harmonious com- 
plexity of the world. To seek not for ends but for “antecedents” 
is the way of the physicist, who finds “‘causes” in what he has 
learned to recognise as fundamental properties, or inseparable 
concomitants, or unchanging laws, of matter and of energy. In 
Aristotle’s parable, the house is there that men may live in it; 
but it is also there because the builders have laid one stone upon 
another: and it is as a mechanism, or a mechanical construction, 
that the physicist looks upon the world. Like warp and woof, 
mechanism and teleology are interwoven together, and we must 
not cleave to the one and despise the other; for their union is 
‘rooted in the very nature of totality§.” 

Nevertheless, when philosophy bids us hearken and obey the 
lessons both of mechanical and of teleological interpretation, the 
precept is hard to follow: so that oftentimes it has come to pass, 
just as in Bacon’s day, that a leaning to the side of the final 
cause “hath intercepted the severe and diligent inquiry of all 

* Alfred Russel Wallace, especially in his later years, relied upon a direct but 
somewhat crude teleology. Cf. his World of Life, a Manifestation of Creative Power, 
Directive Mind and Ultimate Purpose, 1910. 

+ Janet, Les Causes Finales, 1876, p. 350. 

t The phrase is Leibniz’s, in his T'héodicée. 

§ Cf. (int. al.) Bosanquet, The Meaning of Teleology, Proc. Brit. Acad. 
1905-6, pp. 235-245. Cf. also Leibniz (Discours de Métaphysique; Lettres inédites, 
ed. de Careil, 1857, p. 354; cit. Janet, p. 643), “L’un et Vautre est bon, Pun et 


Vautre peut étre utile...et les auteurs qui suivent ces routes différentes ne devraient 
point se maltraiter: et seq.” 


6 INTRODUCTORY [CH. 


real and physical causes,” and has brought it about that “the 
search of the physical cause hath been neglected and passed in 
silence.” So long and so far as “fortuitous variation *” and the 
“survival of the fittest” remain engrained as fundamental and 
satisfactory hypotheses in the philosophy of biology, so long will 
these “satisfactory and specious causes” tend to stay “severe and 
diligent inquiry,” “‘to the great arrest and prejudice of future 
discovery.’ 

The difficulties which surround the concept of active or “real” 
causation, in Bacon’s sense of the word, difficulties of which 
Hume and Locke and Aristotle were little aware, need*scarcely 
hinder us in our physical enquiry. As students of mathematical 
and of empirical physics, we are content to deal with those ante- 
cedents, or concomitants, of our phenomena, without which the 
phenomenon does not occur,—with causes, in short, which, aliae 
ex aliis aptae et necessitate nexae, are no more, and no less, than 
conditions sie quad non. Our purpose is still adequately fulfilled : 
inasmuch as we are still enabled to correlate, and to equate, our 
particular phenomena with more and ever more of the physical 
phenomena around, and so to weave a web of connection and 
interdependence which shall serve our turn, though the meta- 
physician withhold from that interdependence the title of causality. 
We come in touch: with what the schoolmen called a ratio 
cognoscendi, though the true ratio efficiendi is still enwrapped in 
many mysteries. And so handled, the quest of physical causes 
merges with another great Aristotelian theme,—the search for 
relations between things apparently disconnected, and for ‘ simili- 
tude in things to common view unlike.”” Newton did not shew 
the cause of the apple falling, but he shewed a similitude between 
the apple and the stars. 

Moreover, the naturalist and the physicist will continue to 
speak of “causes,” just as of old, though it may be with some 
mental reservations: for, as a French philosopher said, in a 
kindred difficulty: ‘“‘ce sont la des maniéres de s’exprimer, 


* The reader will understand that I speak, not of the “severe and diligent 
inquiry’’ of variation or of “fortuity,” but merely of the easy assumption that 
these phenomena are a sufficient basis on which to rest, with the all-powerful 
help of natural selection, a theory of definite and progressive evolution. 


1] OF TELEOLOGY AND MECHANISM 7 


et si elles sont interdites il faut renoncer a parler de ces 
choses.” 

The search for differences or essential contrasts between the 
phenomena of organic and inorganic, of animate and inanimate 
things has occupied many mens’ minds, while the search for 
community of principles, or essential similitudes, has been followed 
by few; and the contrasts are apt to loom too large, great as 
they may be. M. Dunan, discussing the “Probléme de la Vie*” 
in an essay which M. Bergson greatly commends, declares: “ Les 
lois physico-chimiques sont aveugles et brutales; 1a ot elles 
régnent seules, au lieu d’un ordre et d’un concert, il ne peut y 
avoir qu’incohérence et chaos.’ But the physicist proclaims 
aloud that the physical phenomena which meet us by the way 
have their manifestations of form, not less beautiful and scarce 
less varied than those which move us to admiration among living 
things. The waves of the sea, the little ripples on the shore, the 
sweeping curve of the sandy bay between its headlands, the 
outline of the hills, the shape of the clouds, all these are so many 
riddles of form, so many problems of morphology, and all of 
them the physicist can more or less easily read and adequately 
solve: solving them by reference to their antecedent phenomena, 
in the material system of mechanical forces to which they belong, 
and to which we interpret them as being due. They have also, 
doubtless, their immanent teleological significance; but it is on 
another plane of thought from the physicist’s that we contemplate 
their intrinsic harmony and perfection, and “see that.they are 
good.” 

Nor is it otherwise with the material forms of living things. 
Cell and tissue, shell and bone, leaf and flower, are so many 
portions of matter, and it is in obedience to the laws of physics 
that their particles have been moved, moulded and conformed y. 


* Revue Philosophique. xxxiu, 1892. 

+ This general principle was clearly grasped by Dr George Rainey (a learned 
physician of St Bartholomew’s) many years ago, and expressed in such words 
as the following: “...... it is illogical to suppose that in the case of vital organisms 
a distinct force exists to produce results perfectly within the reach of physical 
agencies, especially as in many instances no end could be attained were that the 
case, but that of opposing one force by another capable of effecting exactly 
the same purpose.” (On Artificial Calculi, Q.J.M.S. (Trans. Microsc. Soc.), V1, 
p. 49, 1858.) Cf. also Helmholtz. infra cit., p. 9. 


8 INTRODUCTORY [on 


They are no exception to the rule that Qeos det yeametpec. Their 
problems of form are in the first instance mathematical problems, 
and their problems of growth are essentially physical problems; 
and the morphologist is, 7pso facto, a student of physical science. 

Apart from the physico-chemical problems of modern physio- 
logy, the road of physico-mathematical or dynamical investigation 
in morphology has had few to follow it; but the pathway is old. 
The way of the old Ionian physicians, of Anaxagoras*, of 
Empedocles and his disciples in the days before Aristotle, lay 
just by that highwayside. It was Galileo’s and Borelli’s way. 
It was little trodden for long afterwards, but once in a while 
Swammerdam and Réaumur looked that way. And of later 
years, Moseley and Meyer, Berthold, Errera and Roux have 
been among the little band of travellers. We need not wonder 
if the way be hard to follow, and if these wayfarers have yet 
gathered little. A harvest has been reaped by others, and the 
gleaning of the grapes is slow. 

It behoves us always to remember that in physics it has taken 
great men to discover simple things. They are very great names 
indeed that we couple with the explanation of the path of a stone, 
the droop of a chain, the tints of a bubble, the shadows in a cup. 
It is but the slightest adumbration of a dynamical morphology 
that we can hope to have, until the physicist and the mathematician 
shall have made these problems of ours their own, or till a new 
Boscovich shall have written for the naturalist the new Theoria 
Philosophiae Naturalis. 

How far, even then, mathematics will suffice to describe, and 
physics to explain, the fabric of the body no man can foresee. 
It may be that all the laws of energy, and all the properties of 
matter, and all the chemistry of all the colloids are as powerless 
to explain the body as they are impotent to comprehend the 
soul. For my part, I think it is not so. Of how it is that the 
soul informs the body, physical science teaches me nothing: 
consciousness is not explained to my comprehension by all the 
nerve-paths and “neurones” of the physiologist; nor do I ask of 
physics how goodness shines in one man’s face, and evil betrays 
itself in another. But of the construction and growth and working 


* Whereby he incurred the reproach of Socrates, in the Phaedo. 


1] OF DYNAMICAL MORPHOLOGY 9 


of the body, as of all that is of the earth earthy, physical science 
is, in my humble opinion, our only teacher and guide * 

Often and often it happens that our physical knowledge is 
inadequate to explain the mechanical working of the organism; 
the phenomena are superlatively complex, the procedure is 
involved and entangled, and the investigation has occupied but 
a few short lives of men. When physical science falls short of 
explaining the order which reigns throughout these manifold 
phenomena,—an order more characteristic in its totality than any 
of its phenomena in themselves,—men hasten to invoke a guiding 
principle, an entelechy, or call it what you will. But all the while, 
so far as I am aware, no physical law, any more than that of 
gravity itself, not even among the puzzles of chemical ‘‘stereo- 
metry,’ or of physiological “surface-action” or “osmosis,” is 
known to be transgressed by the bodily mechanism. 

Some physicists declare, as Maxwell did, that atoms or mole- 
cules more complicated by far than the chemist’s hypotheses 
demand are requisite to explain the phenomena of life. If what 
is implied be an explanation of psychical phenomena, let the 
point be granted at once; we may go yet further, and decline, 
with Maxwell, to believe that anything of the nature of physical 
complexity, however exalted, could ever suffice. Other physicists, 
like Auerbach, or Larmor, or Joly §, assure us that our laws of 
thermodynamics do not suffice, or are “inappropriate,” to explain 
the maintenance or (in Joly’s phrase) the “‘accelerative absorption” 


* In a famous lecture (Conservation of Forces applied to Organic Nature, 
Proc. Roy. Instit.. April 12, 1861), Helmholtz laid it down, as “the fundamental 
principle of physiology,” that “There may be other agents acting in the living 
body than those agents which act in the inorganic world; but those forces, as far 
as they cause chemical and mechanical influence in the body, mustebe quite of the 
same character as inorganic forces: in this at least, that their effects must be ruled 
by necessity, and must always be the same when acting in the same conditions; 
and so there cannot exist any arbitrary choice in the direction of their actions.” 
It would follow from this, that, like the other “physical” forces, they must be 
subject to mathematical analysis and deduction. Cf. also Dr T. Young’s Croonian 
Lecture On the Heart and Arteries, Phil. Trans. 1809, p. 1: Coll. Works, 1, 511. 

+ Ektropismus, oder die physikalische Theorie des Lebens, Leipzig, 1910. 

{~ Wilde Lecture, Nature, March 12, 1908; ibid. Sept. 6, 1900, p. 485; 
Aether and Matter, p. 288. Cf. also Lord Kelvin, Fortnightly Review, 1892, p. 313. 

§ Joly, The Abundance of Life, Proc. Roy. Dublin Soc. vu, 1890; and in 
Scientific Essays. etc. 1915, p. 60 et seq. 


10 INTRODUCTORY [cH. 


of the bodily energies, and the long battle against the cold and 
darkness which is death. With these weighty problems I am not 
for the moment concerned. My sole purpose is to correlate with 
mathematical statement and physical law certain of the simpler 
outward phenomena of organic growth and structure or form: 
while all the while regarding, ex hypothesi, for the purposes of 
this correlation, the fabric of the organism as a material and 
mechanical configuration. 

Physical science and philosophy stand side by side, and one 
upholds the other. Without something of the strength of physics, 
philosophy would be weak; and without something of philosophy’s 
wealth, physical science would be poor. ‘Rien ne retirera du 
tissu de la science les fils d’or que la main du philosophe y a 
introduits*.”” But there are fields where each, for a while at 
least, must work alone; and where physical science reaches its 
hmitations, physical science itself must help us to discover. 
Meanwhile the appropriate and legitimate postulate of the 
physicist, in approaching the physical problems of the body, is 
that with these physical phenomena no alien influence interferes. 
But the postulate, though it is certainly legitimate, and though 
it is the proper and necessary prelude to scientific enquiry, may 
some day be proven to be untrue; and its disproof will not be to 
the physicist’s confusion, but will come as his reward. In dealing 
with forms which are so concomitant with life that they are 
seemingly controlled by life, it is in no spirit of arrogant assertive- 
ness that the physicist begins his argument, after the fashion of 
a most illustrious exemplar, with the old formulary of scholastic 
challenge,—An Vita sit? Dico quod non. 


The terms Form and Growth, which make up the title of this 
little book, aré to be understood, as I need hardly say, in their 
relation to the science of organisms. We want to see how, in 
some cases at least, the forms of living things, and of the parts 
of living things, can be explained by physical considerations, and 
to realise that, in general, no organic forms exist save such as 
are in conformity with ordinary physical laws. And while growth 
is a somewhat vague word for a complex matter, which may 


* Papillon, Histoire de la philosophie moderne, t, p. 300. 


aa OF MATTER AND ENERGY i 


depend on various things, from simple imbibition of water to the 
complicated results of the chemistry of nutrition, it deserves to 
be studied in relation to form, whether it proceed by simple 
increase of size without obvious alteration of form, or whether it 
so proceed as to bring about a gradual change of form and the 
slow development of a more or less complicated structure. 

In the Newtonian language of elementary physics, force is 
recognised by its action in producing or in changing motion, or 
in preventing change of motion or in maintaining rest. When we 
deal with matter in the concrete, force does not, strictly speaking, 
enter into the question, for force, unlike matter, has no independent 
objective existence. It is energy in its various forms, known or 
unknown, that acts upon matter. But when we abstract our 
thoughts from the material to its form, or from the thing moved 
to its motions, when we deal with the subjective conceptions of 
form, or movement, or the movements that change of form implies, 
then force is the appropriate term for our conception of the causes 
by which these forms and changes of form are brought about. 
When we use the term force, we use it, as the physicist always 
does, for the sake of brevity, using a symbol-for the magnitude 
and direction of an action in reference to the symbol or diagram 
of a material thing. It is a term as subjective and symbolic as 
form itself, and so is appropriately to be used in connection 
therewith. 

The form, then, of any portion of matter, whether it be living 
or dead, and the changes of form that are apparent in its movements 
and in its growth, may in all cases alike be described as due to 
the action of force. In short, the form of an object is a “diagram 
of forces,’ in this sense, at least, that from it we can judge of or 
deduce the forces that are acting or have acted upon it: in this 
strict and particular sense, it is a diagram,—in the case of a solid, 
of the forces that have been impressed upon it when its conformation 
was produced, together with those that enable it to retain its 
conformation; in the case of a liquid (or of a gas) of the forces that 
are for the moment acting on it to restrain or balance its own 
inherent mobility. In an organism, great or small, it is not 
merely the nature of the motions of the living substance that we 
must interpret in terms of force (according to kinetics), but also 


12. INTRODUCTORY (oH. 


the conformation of the organism itself, whose permanence or 
equilibrium is explained by the interaction or balance of forces, 
as described in statics. 

If we look at the living cell of an Amoeba or a Spirogyra, we 
see a something which exhibits certain active movements, and 
a certain fluctuating, or more or less lasting, form; and its form 
at a given moment, just like its motions, is to be investigated by 
the help of physical methods, and explained by the invocation of 
the mathematical conception of force. 

Now the state, including the shape or form, of a portion of 
matter, is the resultant of a number of forces, which represent or 
symbolise the manifestations of various kinds of energy; and it 
is obvious, accordingly, that a great part of physical science must 
be understood or taken for granted as the necessary preliminary 
to the discussion on which we are engaged. But we may at 
least try to indicate, very briefly, the nature of the principal 
forces and the principal properties of matter with which our 
subject obliges us to deal. Let us imagine, for instance, the case 
of a so-called “simple” organism, such as Amoeba; and if our 
short list of its physical properties and conditions be helpful 
to our further discussion, we need not consider how far it 
be complete or adequate from the wider physical point of 
view *. 

This portion of matter, then, is kept together by the inter- 
molecular force of cohesion; in the movements of its particles 
relatively to one another, and in its own movements relative to 
adjacent matter, it meets with the opposing force of friction. 
It is acted on by gravity, and this force tends (though slightly, 
owing to the Amoeba’s small mass, and to the small difference 
between its density and that of the surrounding fluid), to flatten. 
it down upon the solid substance on which it may be creeping. 
Our Amoeba tends, in the next place, to be deformed by any 
pressure from outside, even though slight, which may be applied 
to it, and this circumstance shews it to consist of matter in a 
fluid, or at least semi-fluid, state: which state is further indicated 
when we observe streaming or current motions in its interior. 


* With the special and important properties of colloidal matter we are, for 
the time being, not concerned. 


1] OF MATTER AND ENERGY 13 


Like other fluid bodies, its surface, whatsoever other substance, 
gas, liquid or solid, it be in contact with, and in varying degree 
according to the nature of that adjacent substance, is the seat 
of molecular force exhibiting itself as a surface-tension, from the 
action of which many important consequences follow, which 
greatly affect the form of the fluid surface. 

While the protoplasm of the Amoeba reacts to the shghtest 
pressure, and tends to “flow,” and while we therefore speak of it 
as a fluid, it is evidently far less mobile than such a fluid, for 
instance, as water, but is rather like treacle in its slow creeping 
movements as it changes its shape in response to force. Such 
fluids are said to have a high viscosity, and this viscosity obviously 
acts in the way of retarding change of form, or in other words 
of retarding the effects of any disturbing action of force. When 
the viscous fluid is capable of being drawn out into fine threads, 
a property in which we know that the material of some Amoebae 
differs greatly from that of others, we say that the fluid is also 
viscid, or exhibits viscidity. Again, not by virtue of our Amoeba 
being liquid, but at the same time in vastly greater measure than if it 
were a solid (though far less rapidly than if it were a gas), a process 
of molecular diffusion is constantly going on within its substance, 
by which its particles interchange their places within the mass, 
while surrounding fluids, gases and solids in solution diffuse into 
and out of it. In so far as the outer wall of the cell is different 
in character from the interior, whether it be a mere pellicle as 
in Amoeba or a firm cell-wall as in Protococcus, the diffusion 
which takes place through this wall is sometimes distinguished 
under the term osmosis. 

Within the cell, chemical forces are at work, and so also in 
all probability (to judge by analogy) are electrical forces; and 
the organism reacts also to forces from without, that have their 
origin in chemical, electrical and thermal influences. The pro- 
cesses of diffusion and of chemical activity within the cell result, 
by the drawing in of water, salts, and food-material with or without 
chemical transformation into protoplasm, in growth, and this 
complex phenomenon we shall usually, without discussing its 
nature and origin, describe and picture as a force. Indeed we 
shall manifestly be inclined to use the term growth in two senses, 


14 INTRODUCTORY [CH. 


just indeed as we do in the case of attraction or gravitation, 
on the one hand as a process, and on the other hand as a 
force. 

In the phenomena of cell-division, in the attractions or repul- 
sions of the parts of the dividing nucleus and in the “ caryokinetic ” 
figures that appear in connection with it, we seem to see in opera- 
tion forces and the effects of forces, that have, to say the least of 
it, a close analogy with known physical phenomena; and to this 
matter we shall afterwards recur. But though they resemble 
known physical phenomena, their nature is still the subject of 
much discussion, and neither the forms produced nor the forces 
at work can yet be satisfactorily and simply explained. We may 
readily admit, then, that besides phenomena which are obviously 
physical in their nature, there are actions visible as well as 
invisible taking place within living cells which our knowledge 
does not permit us to ascribe with certainty to any known physical 
force; and it may or may not be that these phenomena will yield 
in time to the methods of physical investigation. Whether or 
no, it is plain that we have no clear rule or guide as to what is 
“vital”? and what is not; the whole assemblage of so-called vital 
phenomena, or properties of the organism, cannot be clearly 
classified into those that are physical in origin and those that are 
sui generis and peculiar to living things. All we can do meanwhile 
is to analyse, bit by bit, those parts of the whole to which the 
ordinary laws of the physical forces more or less obviously and 
clearly and indubitably apply. 

Morphology then is not only a study of material things and 
of the forms of material things, but has its dynamical aspect, 
under which we deal with the interpretation, in terms of force, 
of the operations of Energy. And here it is well worth while 
to remark that, in dealing with the facts of embryology or the 
phenomena of inheritance, the common language of the books 
seems to deal too much with the material elements concerned, as 
the causes of development, of variation or of hereditary trans- 
mission. Matter as such produces nothing, changes nothing, does 
nothing; and however convenient it may afterwards be to abbre- 
viate our nomenclature and our descriptions, we must most 
carefully realise in the outset that the spermatozoon, the nucleus, 


1] OF MATTER AND ENERGY 15 


the chromosomes or the germ-plasm can never act as matter 
alone, but only as seats of energy and as centres of force. And 
this is but an adaptation (in the light, or rather in the con- 
ventional symbolism, of modern physical science) of the old 
saying of the philosopher: apyy yap % divas wadrov THs Orns. 


CHAPTER II 
ON MAGNITUDE 


To terms of magnitude, and of direction, must we refer all 
our conceptions of form. For the form of an object is defined 
when ‘we know its magnitude, actual or relative, in various 
directions; and growth involves the same conceptions of magnitude 
and direction, with this addition, that they are supposed to alter 
in time. Before we proceed to the consideration of specific form, 
it will be worth our while to consider, for a little while, certain 
phenomena of spatial magnitude, or of the extension of a body 
in the several dimensions of space*. 

We are taught by elementary mathematics that, in similar 
solid figures, the surface increases as the square, and the volume 
as the cube, of the linear dimensions. If we take the simple case 
of a sphere, with radius 7, the area of its surface is equal to 4z7?, 
and its volume to 4a7*; from which it follows that the ratio of 
volume to surface, or V/S, is 47. In other words, the greater the 
radius (or the larger the sphere) the greater will be its volume, or 
its mass (if it be uniformly dense throughout), in comparison with 
its superficial area. And, taking L to represent any linear dimen- 
sion, we may write the general equations in the form 


recuese Voc 8. 


or S ==. > and V =k. ie 
and ae Ti 


From these elementary principles a great number of conse- 
quences follow, all more or less interesting, and some of them of 
great importance. In the first place, though growth in length (let 


* Cf. Hans Przibram, Anwendung elementarer Mathematik auf Buologische 
Probleme (in Roux’s Vortrdge, Heft 1), Leipzig, 1908, p. 10. 


CH. IT] OF SURFACE AND VOLUME 17 


us say) and growth in volume (which is usually tantamount to 
mass or weight) are parts of one and the same process or pheno- 
menon, the one attracts our attention by its increase, very much 
more than the other. . For instance a fish, in doubling its length, 
multiplies its weight by no less than eight times; and it all but 
doubles its weight in growing from four inches long to five. 

In the second place we see that a knowledge of the correlation 
between length and weight in any particular species of animal, 
in other words a determination of & in the formula W =k. L’, 
enables us at any time to translate the one magnitude into the 
other, and (so to speak) to weigh the animal with a measuring- 
rod; this however being always subject to the condition that the 
animal shall in no way have altered its form, nor its specific 
gravity. That its specific gravity or density should materially or 
rapidly alter is not very likely; but as long as growth lasts, 
changes of form, even though inappreciable to the eye, are likely 
to go on. Now weighing is a far easier and far more accurate 
operation than measuring; and the measurements which would 
reveal slight and otherwise imperceptible changes in the form of 
a fish—shght relative differences between, length, breadth. and 
depth, for instance,—would need to be very delicate indeed. But 
if we can make fairly accurate determinations of the length, 
which is very much the easiest dimension to measure, and then 
correlate it with the weight, then the value of k, according to 
whether it varies or remains constant, will tell us at once whether 
there has or has not been a tendency to gradual alteration in the 
general form, To this subject we shall return, when we come to 
consider more particularly the rate of growth. 

But a much deeper interest arises out of this changing ratio 
of dimensions when we come to consider the inevitable changes 
of physical relations with which it is bound up. We are apt, and 
even accustomed, to think that magnitude is so purely relative 
that differences of magnitude make no other or more essential 
difference; that Lilliput and Brobdingnag are all alike, according 
as we look at them through one end of the glass or the other. 
But this is by no means so; for scale has a very marked effect 
upon physical phenomena, and the effect of scale constitutes what 
is known as the principle of similitude, or of dynamical similarity. 


mT Ge Z 


18 ON MAGNITUDE [CH. 


This effect of scale is simply due to the fact that, of the physical 
forces, some act either directly at the surface of a body, or other- 
Wise in proportion to the area of surface; and others, such as 
gravity, act on all particles, internal and external alike, and exert 
a force which is proportional to the mass, and so usually to the 
volume, of the body. 

The strength of an iron girder obviously varies with the 
cross-section of its members, and each cross-section varies as the 
square of a linear dimension; but the weight of the whole structure 
varies as the cube of its linear dimensions. And it follows at once 
that, if we build two bridges geometrically similar, the larger is 
the weaker of the two*. It was elementary engineering experience 
such as this that led Herbert Spencer 7 to apply the principle of 
similitude to biology. 

The same principle had been admirably applied, in a few clear 
instances, by Lesage t, a celebrated eighteenth century physician 
of Geneva, in an unfinished and unpublished work§. Lesage 
argued, for instance, that the larger ratio of surface to mass would 
lead in a small animal to excessive transpiration, were the skin 
as “porous” as our own; and that we may hence account for 
the hardened or thickened skins of insects and other small terrestrial 
animals. Again, since the weight of a fruit increases as the cube 
of its dimensions, while the strength of the stalk increases as the 
square, it follows that the stalk should grow out of apparent due 
proportion to the fruit; or alternatively, that tall trees should 
not bear large fruit on slender branches, and that melons and 
pumpkins must lie upon the ground. And again, that in quad- 
rupeds a large head must be supported on a neck which is either 


* The subject is treated from an engineering point of view by Prof. James 
Thomson, Comparisons of Similar Structures as to Elasticity, Strength, and 
Stability, Trans. Inst. Engineers, Scotland, 1876 (Collected Papers, 1912, pp. 361— 
372), and by Prof. A. Barr, ibid. 1899; see also Rayleigh, Nature, April 22, 1915. 

+ Cf. Spencer, The Form of the Earth, etc., Phil. Mag. xxx, pp. 194-6, 
1847; also Principles of Biology, pt. 1, ch. 1, 1864 (p. 123, etce.). 

+t George Louis Lesage (1724-1803), well known as the author of one of the few 
attempts to explain gravitation. (Cf. Leray, Constitution de la Matiére, 1869; 
Kelvin, Proc. R. S. H. vu, p. 577, 1872, etc.; Clerk Maxwell, Phil. Trans. vol. 157, 
p- 50, 1867; art. “Atom,” Hncycl. Brit. 1875, p. 46.) 

§ Cf. Pierre Prévost, Notices de la vie et des écrits de Lesage, 1805; quoted by 
Janet, Causes Finales, app. 11. 


11] THE PRINCIPLE OF SIMILITUDE 19 


excessively thick and strong, like a bull’s, or very short like the 
neck of an elephant. 

But it was Galileo who, wellnigh 300 years ago, had first laid 
down this general principle which we now know by the name of the 
principle of similitude; and he did so with the utmost possible 
clearness, and with a great wealth of illustration, drawn from 
structures living and dead*. He showed that neither can man 
build a house nor can nature construct an animal beyond a certain 
size, while retaining the same proportions and employing the 
same materials as sufficed in the case of a smaller structureft. 
The thing will fall to pieces of its own weight unless we either 
change its relative proportions, which will at length cause it to 
become clumsy, monstrous and inefficient, or else we must find 
a new material, harder and stronger than was used before. Both 
processes are familiar to us in nature and in art, and practical 
applications, undreamed of by Galileo, meet us at every turn in 
this modern age of steel. 

Again, as Galileo was also careful to explain, besides the 
questions of pure stress and strain, of the strength of muscles to 
lift an increasing weight or of bones to resist its crushing stress, 
we have the very important question of bending moments. This 
question enters, more or less, into our whole range of problems; 
it affects, as we shall afterwards see, or even determines the whole 
form of the skeleton, and is very important in such a case as that 
of a tall treet. 

Here we have to determine the poimt at which the tree will 
curve under its own weight, if it be ever so little displaced from 
the perpendicular§. In such an investigation we have to make 


* Discorsi e Dimostrazioni matematiche, intorno a due nuove scienze, 
attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli Elzevirii, mpcxxxvm. 
Opere, ed. Favaro, vu, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 
1914, p. 130, etc. See Nature, June 17, 1915. 

+ So Werner remarked that Michael Angelo and Bramanti could not have built 
of gypsum at Paris on the scale they built of travertin in Rome. 

{ Sir G. Greenhill, Determination of the. greatest height to which a Tree of 
given proportions can grow, Cambr. Phil. Soc. Pr. tv, p. 65, 1881, and Chree, 
ibid. vit, 1892. Cf. Poynting and Thomson’s Properties of Matter, 1907, p 99. 

§ In like manner the wheat-straw bends over under the weight of the loaded 
ear, and the tip of the cat’s tail bends over when held upright,—not because they 
“possess flexibility,” but because they outstrip the dimensions within which stable 


F—9 


20 ON MAGNITUDE [CH. 


some assumptions,—for instance, with regard to the trunk, that 
it tapers uniformly, and with regard to the branches that their 
sectional area varies according to some definite law, or (as Ruskin 
assumed*) tends to be constant in any horizontal plane; and the 
mathematical treatment is apt to be somewhat difficult. But 
Greenhill has shewn that (on such assumptions as the above), 
a certain British Columbian pine-tree, which yielded the Kew flag- 
staff measuring 221 ft. in height with a diameter at the base of 
21 inches, could not possibly, by theory, have grown to more 
than about 300 ft. It is very curious that Galileo suggested 
precisely the same height (dugento braccia alta) as the utmost 
limit of the growth of a tree. In general, as Greenhill shews, the 
diameter of a homogeneous body must increase as the power 3/2 
of the height, which accounts for the slender proportions of young 
trees, compared with the stunted appearance of old and large 
ones}. In short, as Goethe says in Wahrheit und Dichtung, “Es 
ist dafiir gesorgt dass die Baume nicht in den Himmel wachsen.” 
But LHiffel’s great tree of steel (1000 feet high) is built to a 
very different plan; for here the profile of the tower follows the 
logarithmic curve, giving equal strength throughout, according 
to a principle ‘which we shall have occasion to discuss when we 
come to treat of “form and mechanical efficiency” in connection 
with the skeletons of animals. 

Among animals, we may see in a general way, without the help 
of mathematics or of physics, that exaggerated bulk brings with 
it a certain clumsiness, a certain inefficiency, a new element of 
risk and hazard, a vague preponderance of disadvantage. The 
case was well put by Owen, in a passage which has an interest 
of its own as a premonition (somewhat like De Candolle’s) of the 
“struggle for existence.” Owen wrote as followst: “In pro- 
portion to the bulk of a species is the difficulty of the contest 
which, as a living organised whole, the individual of such species 


equilibrium is possible in a vertical position. The kitten’s tail, on the other hand, 
stands up spiky and straight. 

* Modern Painters. 

+ The stem of the giant bamboo may attain a height of 60 metres, while not 
more than about 40 cm. in diameter near its base, which dimensions are not very 
far short of the theoretical limits (A. J. Ewart, Phil. Trans. vol. 198, p. 71, 1906). 

t Trans. Zool. Soc. tv, 1850, p. 27. 


ir] THE PRINCIPLE OF SIMILITUDE aoe 


has to maintain against the surrounding agencies that are ever 
tending to dissolve the vital bond, and subjugate the living 
matter to the ordinary chemical and physical forces. Any 
changes, therefore, in such external conditions as a species may 
have been originally adapted to exist in, will militate against that 
existence in a degree proportionate, perhaps in a geometrical ratio, 
to the bulk of the species. If a dry season be greatly prolonged, 
the large mammal will suffer from the drought sooner than the 
small one; if any alteration of climate affect the quantity of 
vegetable food, the bulky Herbivore will first feel the effects of 
stinted nourishment.” 

But the principle of Galileo carries us much further and along 
more certain lines. 

The tensile strength of a muscle, like that of a rope or of our 
girder, varies with its cross-section; and the resistance of a bone 
to a crushing stress varies, again like our girder, with its cross- 
section. But in a terrestrial animal the weight which tends to 
crush its limbs or which its muscles have to move, varies as the 
cube of its linear dimensions; and so, to the possible magnitude 
of an animal, living under the direct action of gravity, there is 
a definite limit set. The elephant, in the dimensions of its limb- 
bones, is already shewing signs of a tendency to disproportionate 
thickness as compared with the smaller mammals; its movements 
are in many ways hampered and its agility diminished: it is 
already tending towards the maximal limit of size which the 
physical forces permit. But, as Galileo also saw, if the animal 
be wholly immersed in water, like the whale, (or if it be partly 
so, as was in all probability the case with the giant reptiles of our 
secondary rocks), then the weight is counterpoised to the extent 
of an equivalent volume of water, and is completely counterpoised 
if the density of the animal’s body, with the included air, be 
identical (as in a whale it very nearly is) with the water around. 
Under these circumstances there is no longer a physical barrier 
to the indefinite growth in magnitude of the animal*. Indeed, 


* It would seem to be a common if not a general rule that marine organisms, 
zoophytes, molluscs, etc., tend to be larger than the corresponding and closely 
related forms living in fresh water. While the phenomenon may have various 
causes, it has been attributed (among others) to the simple fact that the forces of 
growth are less antagonised by gravity in the denser medium (cf. Houssay, La 


22 ON MAGNITUDE [CH. 


in the case of the aquatic animal there is, as Spencer pointed out, 
a distinct advantage, in that the larger it grows the greater is 
its velocity. For its available energy depends on the mass of 
its muscles; while its motion through the water is opposed, not 
by gravity, but by “skin-friction,’” which increases only as the 
square of its dimensions; all other things being equal, the bigger 
the ship, or the bigger the fish, the faster it tends to go, but only 
in the ratio of the square root of the increasing length. For the 
mechanical work (W) of which the fish is capable being pro- 
portional to the mass of its muscles, or the cube of its linear 
dimensions: and again this work being wholly done in producing 
a velocity (V) against a resistance (#) which increases as the 
square of the said linear dimensions; we have at once 


W = 
and also Why ey 
Therefore 2 PV 2; and We a7 


This is what is known as Froude’s Law of the correspondence of 
speeds. 

But there is often another side to these questions, which makes 
them too complicated to answer in a word. For instance, the 
work (per stroke) of which two similar engines are capable should 
obviously vary as the cubes of their hnear dimensions, for it 
varies on the one hand with the surface of the piston, and on the 
other, with the length of the stroke; so is it likewise in the animal, 
where the corresponding variation depends on the cross-section of 
the muscle, and on the space through which it contracts. But 
in two precisely similar engines, the actual available horse-power 
varies as the square of the linear dimensions, and not as the 
cube; and this for the obvious reason that the actual energy 
developed depends upon the heating-surface of the boiler*. So 
likewise must there be a similar tendency, among animals, for the 
rate of supply of kinetic energy to vary with the surface of the 


Forme et la Vie, 1900, p. 815). The effect of gravity on outward form is 
iilustrated, for instance, by the contrast between the uniformly upward branching 
of a sea-weed and the drooping curves of a shrub or tree. 

* The analogy is not a very strict one. We are not taking account, for instance, 
of a proportionate increase in thickness of the boiler-plates. 


1] OF FROUDE’S LAW 23 


lung, that is to say (other things being equal) with the square of 
the linear dimensions of the animal. We may of course (departing 
from the condition of similarity) increase the heating-surface of 
the boiler, by means of an internal system of tubes, without 
increasing its outward dimensions, and in this very way nature 
increases the respiratory surface of a lung by a complex system 
of branching tubes and minute air-cells; but nevertheless in 
two similar and closely related animals, as also in two steam- 
engines of precisely the same make, the law is bound to hold that 
the rate of working must tend to vary with the square of the 
linear dimensions, according to Froude’s law of steamship com- 
parison. In the case of a very large ship, built for speed, the 
difficulty is got over by increasing the size and number of the 
boilers, till the ratio between boiler-room and engine-room is 
far beyond what is required in an ordinary small vessel*; but 
though we find lung-spaces increased among animals where 
greater rate of working is required, as in general among birds, 
I do not know that it can be shewn to increase, as in the 
“over-boilered” ship, with the size of the animal, and in a ratio 
which outstrips that of the other bodily dimensions. If it be the 
case then, that the working mechanism of the muscles should be 
able to exert a force proportionate to the cube of the linear 
bodily dimensions, while the respiratory mechanism can only 
supply a store of energy at a rate proportional to the square of 
the said dimensions, the singular result ought to follow that, in 
swimming for instance, the larger fish ought to be able to put on 
a spurt of speed far in excess of the smaller one; but the distance 
travelled by the year’s end should be very much alike for both 
of them. And it should also follow that the curve of fatigue 


* Let L be the length, S the (wetted) surface, 7 the tonnage, D the displacement 
(or volume) of a ship; and let it cross the Atlantic at a speed V. Then, in com- 
paring two ships, similarly constructed but of different magnitudes, we know that 
L=V?, S=L?=V1, D=T=L3=Y*; also _R (resistance) —S.V2=V*; H (horse- 
power) = R.V=V’; and the coal (C) necessary for the voyage= H/V=V®, That 
is to say, in ordinary engineering language, to increase the speed across the Atlantic 
by 1 per cent. the ship’s length must be increased 2 per cent., her tonnage or 
displacement 6 per cent., her coal-consumpt also 6 per cent., her horse-power, 
and therefore her boiler-capacity, 7 per cent. Her bunkers, accordingly, keep 
pace with the enlargement of the ship, but her boilers tend to increase out of 
proportion to the space available. 


24 ON MAGNITUDE [CH. 


should be a steeper one, and the staying power should be less, in 
the smaller than in the larger individual. This is the case of long- 
distance racing, where the big winner puts on his big spurt at the 
end. And for an analogous reason, wise men know that in the 
’Varsity boat-race it is judicious and prudent to bet on the heavier 
crew. 

Leaving aside the question of the supply of energy, and keeping 
to that of the mechanical efficiency of the machine, we may find 
endless biological illustrations of the principle of similitude. 

In the case of the flying bird (apart from the initial difficulty of 
raising itself into the air, which involves another problem) it may 
be shewn that the bigger it gets (all its proportions remaining the 
same) the more difficult it is for it to maintain itself aloft in flight. 
The argument is as follows: — 

In order to keep aloft, the bird must communicate to the air 
a downward momentum equivalent: to its own weight, and there- 
fore proportional to the cube of its own linear dimensions. But 
the momentum so communicated is proportional to the mass of 
air driven downwards, and to the rate at which it is driven: the 
mass being proportional to the bird’s wing-area, and also (with 
any given slope of wing) to the speed of the bird, and the rate 
being again proportional to the bird’s speed; accordingly the 
whole momentum varies as the wing-area, i.e. as the square of the 
linear dimensions, and also as the square of the speed. Therefore, 
in order that the bird may maintain level flight, its speed must 
be proportional to the square root of its linear dimensions. 

Now the rate at which the bird, in steady flight, has to work 
in order to drive itself forward, is the rate at which it communicates 
energy to the air; and this is proportional to mV2, i.e. to the 
mass and to the square of the velocity of the air displaced. But 
_the mass of air displaced per second is proportional to the wing- 
area and to the speed of the bird’s motion, and therefore to the 
power 23 of the linear dimensions; and the speed at which it 
is displaced is proportional to the bird’s speed, and therefore to 
the square root of the linear dimensions. Therefore the energy 
communicated per second (being proportional to the mass and to 
the square of the speed) is jointly proportional to the power 24 of 
the linear dimensions, as above, and to the first power thereof : 


Or 


it] THE PRINCIPLE OF SIMILITUDE | 2k 


that is to say, it increases in proportion to the power 34 of the 
linear dimensions, and therefore faster than the weight of the 
bird increases. 

Put in mathematical form, the equations are as follows: 

(m=the mass of air thrust downwards; V its velocity, 
proportional to that of the bird; M its momentum; / a linear 
dimension of the bird; w its weight; W the work done in moving 
itself forward.) 


M=w=P. 
But M=mV, and m=[YS. 
Therefore M=PYV2, 
ee alee 
or V =U. 
But, again, W=mv?2=PV x V2 
== iP Xt a/ lal 
= 1%, 


The work requiring to be done, then, varies as the power 34 of 
the bird’s linear dimensions, while the work of which the bird is 
capable depends on the mass of its muscles, and therefore varies 
as the cube of its linear dimensions*. The disproportion does not 
seem at first sight very great, but it is quite enough to tell. It is 
as much as to say that, every time we double the linear dimensions 
of the bird, the difficulty of flight is increased in the ratio of 
23: 232, or 8: 11-3, or, say,1:1-4. If we take the ostrich to 
exceed the sparrow in linear dimensions as 25 : 1, which seems well 
within the mark, we have the ratio between 25%" and 25%, or 
between 57:58; in other words, flight is just five times more 
difficult for the larger than for the smaller bird ft. 

The above investigation includes, besides the final result, a 
number of others, explicit or implied, which are of not less im- 
portance. Of these the simplest and also the most important is 


* This is the result arrived at by Helmholtz, Ueber ein Theorem geometrisch 
ahnliche Bewegungen fliissiger K6rper betreffend, nebst Anwendung auf das 
Problem Luftballons zu lenken, Monatsber. Akad. Berlin, 1873, pp. 501-14. It 
was criticised and challenged (somewhat rashly) by K. Miillenhof, Die Grosse 
der Flugflaichen, etc., Pfliiger’s Archiv, xxxv, p. 407, xxxvi, p. 548, 1885. 

+ Cf. also Chabrier, Vol des Insectes, Mém. Mus. Hist. Nat. Paris, vi—vi, 
1820-22. 


26 ON MAGNITUDE (cH. 


contained in the equation V = 4//, a result which happens to be 
identical with one we had also arrived at in the case of the fish. 
In the bird’s case it has a deeper significance than in the other; 
because it implies here not merely that the velocity will tend to 
increase in a certain ratio with the length, but that it must do so 
as an essential and primary condition of the bird’s remaining aloft. 
It is accordingly of great practical importance in aeronautics, for 
it shews how a provision of increasing speed must accompany every 
enlargement of our aeroplanes. If a given machine weighing, say, 
500 Ibs. be stable at 40 miles an hour, then one geometrically 
similar which weighs, say, a couple of tons must have its speed 
determined as follows: 


W ep se Pes oe an. 


Therefore — | | Was aap 
But Ve rea 
Therefore Viettna/2 ol = ae A: 


That is to say, the larger machine must be capable of a speed 
equal to 1-414 = 40, or about 563 miles per hour. 

It is highly probable, as Lanchester* remarks, that Lilienthal 
met his untimely death not so much from any intrinsic fault in 
the design or construction of his machine, but simply because his 
engine fell somewhat short of the power required to give the 
speed which was necessary for stability. An arrow is a very 
imperfectly designed aeroplane, but nevertheless it is evidently 
capable, to a certain extent and at a high velocity, of acquiring 
“stability” and hence of actual “flight”: the duration and 
consequent range of its trajectory, as compared with a bullet of 
similar initial velocity, being correspondingly benefited. When 
we return to our birds, and again compare the ostrich with the 
sparrow, we know little or nothing about the speed in flight of 
the latter, but that of the swift is estimated} to vary from a 
minimum of 20 to 50 feet or more per second,—say from 14 to 
35 miles per hour. Let us take the same lower limit as not far 
from the minimal velocity of the sparrow’s flight also; and it 


* Aerial Flight, vol. 1 (Aerodonetics), 1908, p. 150. 
+ By Lanchester, op. cit. p. 131. 


11] THE PRINCIPLE OF SIMILITUDE 27 


would follow that the ostrich, of 25 times the sparrow’s linear 
. dimensions, would be compelled to fly (if it flew at all) with 
a minimum velocity of 5 x 14, or 70 miles an hour. 

The same principle of necessary speed, or the indispensable 
relation between the dimensions of a flying object and the minimum 
velocity at which it is stable, accounts for a great number of 
observed phenomena. It tells us why the larger birds have a 
marked difficulty in rising from the ground, that is to say, in 
acquiring to begin with the horizontal velocity necessary for their 
support; and why accordingly, as Mouillard* and others have 
observed, the heavier birds, even those weighing no more than 
a pound or two, can be effectively “caged” in a small enclosure 
open to the sky. It tells us why very small birds, especially 
those as small as humming-birds, and a@ fortiori the still smaller 
insects, are capable of “stationary flight,’ a very slight and 
scarcely perceptible velocity relatively to the air being sufficient for 
their support and stability. And again, since it is in all cases 
velocity relative to the air that we are speaking of, we comprehend 
the reason why one may always tell which way the wind blows 
by watching the direction in which a bird starts to fly. 

It is not improbable that the ostrich has already reached 
a magnitude, and we may take it for certain that the moa did 
so, at which flight by muscular action, according to the normal 
anatomy of a bird, has become physiologically impossible. The 
Same reasoning applies to the case of man. It would be very 
difficult, and probably absolutely impossible, for a bird to fly 
were it the bigness of a man. But Borelli, in discussing this 
question, laid even greater stress on the obvious fact that a man’s 
pectoral muscles are so immensely less in proportion than those 
of a bird, that however we may fit ourselves with wings we can 
never expect to move them by any power of our own relatively 
weaker muscles; so it is that artificial flight only became possible 
when an engine was devised whose efficiency was extraordinarily 
great in comparison with its weight and size. 

Had Leonardo da Vinci known what Galileo knew, he would 
not have spent a great part of his life on vain efforts to make to 
himself wings. Borelli had learned the lesson thoroughly, and 


* Cf. empire de Vair ; ornithologie appliquée a Paviation. 1881. 


28 ON MAGNITUDE [CH. 


in one of his chapters he deals with the proposition, “Est im- 
possibile, ut homines propriis viribus artificiose volare possint*.” 

But just as it is easier to swim than to fly, so is it obvious 
that, in a denser atmosphere, the conditions of flight would be 
altered, and flight facilitated. We know that in the carboniferous 
epoch there lived giant dragon-flies, with wings of a span far 
greater than nowadays they ever attain; and the small bodies 
and huge extended wings of the fossil pterodactyles would seem 
in like manner to be quite abnormal according to our present 
standards, and to be beyond the limits of mechanical efficiency 
under present conditions. But as Harlé suggestst, following 
upon a suggestion of Arrhenius, we have only to suppose that in 
carboniferous and jurassic days the terrestrial atmosphere was 
notably denser than it is at present, by reason, for instance, of 
its containing a much larger proportion of carbonic acid, and we 
have at once a means of reconciling the apparent mechanical 
discrepancy. 

Very similar problems, involving in various ways the principle 
of dynamical similitude, occur all through the physiology of 
locomotion: as, for instance, when we see that a cockchafer can 
carry a plate, many times his own weight, upon his back, or that 
a flea can jump many inches high. 

Problems of this latter class have been admirably treated both 
by Galileo and by Borelli, but many later writers have remained 
ignorant of their work. Linnaeus, for instance, remarked that, 
if an elephant were as strong in proportion as a stag-beetle, it 
would be able to pull up rocks by the root, and to level mountains. 
And Kirby and Spence have a well-known passage directed to 
shew that such powers as have been conferred upon the insect 
have been withheld from the higher animals, for the reason that 
had these latter been endued therewith they would have “caused 
the early desolation of the world{.” 


* De Motu Animalium, I, prop. cciv, ed. 1685, p. 243. 

+ Harlé, On Atmospheric Pressure in past Geological Ages, Bull. Geol. Soc. 
Fr, x1, pp. 118-121; or Cosmos, p. 30, July 8, 1911. 

t Introduction to Entomology, 1826, 1, p. 190. K. and §., like many less learned 
authors, are fond of popular illustrations of the ‘‘ wonders of Nature,” to the neglect 
of dynamical principles. They suggest, for instance, that if the white ant were 
as big as a man, its tunnels would be ‘“‘magnificent cylinders of more than three 


11] BORELLI?’S LAW 29 


Such problems as that which is presented by the flea’s jumping 
powers, though essentially physiological in their nature, have their 
interest for us here: because a steady, progressive diminution of 
activity with increasing size would tend to set limits to the possible 
growth in magnitude of an animal just as surely as those factors 
which tend to break and crush the living fabric under its own 
weight. In the case of a leap, we have to do rather with a sudden 
impulse than with a continued strain, and this impulse should be 
measured in terms of the velocity imparted. The velocity is 
proportional to the impulse (z), and inversely proportional to the 
mass (M) moved: V=2/M. But, according to what we still speak 
of as “‘ Borelli’s law,” the impulse (i.e. the work of the impulse) is 
proportional to the volume of the muscle by which it is produced*, 
that is to say (in similarly constructed animals) to the mass of the 
whole body; for the impulse is proportional on the one hand to 
the cross-section of the muscle, and on the other to the distance 
through which it contracts. It follows at once from this that the 
_ velocity is constant, whatever be the size of the animals: in 
other words, that all animals, provided always that they are 
similarly fashioned, with their various levers etc., in like proportion, 
ought to jump, not to the same relative, but to the same actual 
heightt. According to this, then, the flea is not a better, but 
rather a worse jumper than a horse or a man. As a matter of 
fact, Borelli is careful to point out that in the act of leaping the 
impulse is not actually instantaneous, as in the blow of a hammer, 
but takes some little time, during which the levers are being 
extended by which the centre of gravity of the animal is being 
propelled forwards; and this interval of time will be longer in 
the case of the longer levers of the larger animal. To some extent, 
then, this principle acts as a corrective to the more general one, 
hundred feet in diameter”; and that if a certain noisy Brazilian insect were as 
big as a man, its voice would be heard all the world over: “so that Stentor 
becomes a mute when compared with these insects!” It is an easy consequence 
of anthropomorphism, and hence a common characteristic of fairy-tales, to neglect 
the principle of dynamical, while dwelling on the aspect of geometrical, similarity. 

* T.e. the available energy of muscle, in ft.-lbs. per lb. of muscle, is the same 
for all animals: a postulate which requires considerable qualification when we are 
comparing very different kinds of muscle, such as the insect’s and the mammal’s. 


+ Prop. elxxvii. Animalia minora et minus ponderosa majores saltus efficiunt 
respectu sui corporis, si caetera fuerint paria. 


30 ON MAGNITUDE © [cH. 


and tends to leave a certain balance of advantage, in regard to 
leaping power, on the side of the larger animal*. 

But on the other hand, the question of strength of materials 
comes in once more, and the factors of stress and strain and 
bending moment make it, so to speak, more and more difficult 
for nature to endow the larger animal with the length of lever 
with which she has provided the flea or the grasshopper. 

To Kirby and Spence it seemed that “This wonderful strength 
of insects is doubtless the result of something peculiar in the 
structure and arrangement of their muscles, and principally their 
extraordinary power of contraction.” This hypothesis, which is 
so easily seen, on physical grounds, to be unnecessary, has been 
amply disproved in a series of excellent papers by F. Plateauy. 

A somewhat simple problem is presented to us by the act of 
walking. It is obvious that there will be a great economy of 
work, if the leg swing at its normal pendulum-rate; and, though 
this rate is hard to calculate, owing to the shape and the jointing 
of the limb, we may easily convince ourselves, by counting our 
steps, that the leg does actually swing, or tend to swing, just as 
a pendulum does, at a certain definite ratet. When we walk 
quicker, we cause the leg-pendulum to describe a greater arc, but 
we do not appreciably cause it to swing, or vibrate, quicker, until 
we shorten the pendulum and begin to run. Now let two indi- 
viduals, A and B, walk in a similar fashion, that is to say, with 
a similar angle of swing. The arc through which the leg swings, 
or the amplitude of each step, will therefore vary as the length 
of leg, or say as a/b; but the time of swing will vary as the square 

* See also (int. al.), John Bernoulli, de Motu Musculorwm, Basil., 1694; 
Chabry, Mécanisme du Saut, J. de Anat. et de la Physiol. x1x, 1883; Sur la’ 
longueur des membres des animaux sauteurs, ibid. xxi, p. 356, 1885; Le Hello, 
De Vaction des organes locomoteurs, etc., ibid. XXIx, p. 65-93, 1893, etc. 

y+ Recherches sur la force absolue des muscles des Invertébrés, Bull. Acad. R. 
de Belgique (3), VI, Vu, 1883-84: see also ibid. (2), xx, 1865, xx, 1866; Ann. 
Mag. N. H. xvu, p. 139, 1866, xix, p. 95, 1867. The subject was also well treated 
by Straus-Diirckheim, in his Considérations générales sur Vanatomie comparée des 
animaux articulés, 1828. 

t The fact that the limb tends to swing in pendulum-time was first observed 
by the brothers Weber (Mechanik der menschl. Gehwerkzeuge, Gottingen, 1836). 
Some later writers have criticised the statement (e.g. Fischer, Die Kinematik des 


Beinschwingens etc., Abh. math. phys. Kl. k. Sachs. Ges. xxv—xxvut, 1899-1903), 
but for all that, with proper qualifications, it remains substantially true. 


11] THE PRINCIPLE OF SIMILITUDE 31 


root of the pendulum-length, or \/a/1/b. Therefore the velocity, 
which is measured by ae will also vary as the square- 
roots of the length of leg: that is to say, the average velocities of 
A and B are in the ratio of \/a: +/b. 

The smaller man, or smaller animal, is so far at a disadvantage 
compared with the larger in speed, but only to the extent of the 
ratio between the square roots of their linear dimensions: whereas, 
if the rate of movement of the limb were identical, irrespective 
of the size of the animal,—if the limbs of the mouse for instance 
swung at the same rate as those of the horse,—then, as F. Plateau 
said, the mouse would be as slow or slower in its gait than the 
tortoise. M. Delisle* observed a “minute fly” walk three inches 
in half-a-second. This was good steady walking. When we 
walk five miles an hour we go about 88 inches in a second, or 
_ 88/6 = 14-7 times the pace of M. Delisle’s fly. We should walk 
at just about the fly’s pace if our stature were 1/(14-7)?, or 1/216 

of our present height,—say 72/216 inches, or one-third of an inch 
high. , : 
But the leg comprises a complicated system of levers, by whose 
various exercise we shall obtain very different results. For 
instance, by being careful to rise upon our instep, we considerably 
increase the length or amplitude of our stride, and very considerably 
increase our speed accordingly. On the other hand, in running, 
we bend and so shorten the leg, in order to accommodate it to 
a quicker rate of pendulum-swingt. In short, the jointed structure 
of the leg permits us to use it as the shortest possible pendulum 
when it is swinging, and as the longest possible lever when it is 
exerting its propulsive force. 

Apart from such modifications as that described in the last 
paragraph,—apart, that is to say, from differences in mechanical 
construction or in the manner in which the mechanism is used,— 
we have now arrived at a curiously simple and uniform result. 
For in all the three forms of locomotion which we have attempted 


* Quoted in Mr John Bishop’s interesting article in Todd’s Cyclopaedia, ut, 
p. 443. 

{ There is probably also another factor involved here: for in bending, and there- 
fore shortening, the leg we bring its centre of gravity nearer to the pivot, that is 
to say, to the joint, and so the muscle tends to move it the more quickly. 


32 ON MAGNITUDE “[ox. 


to study, alike in swimming, in flight and in walking, the general 
result, attained under very different conditions and arrived at by | 
very different modes of reasoning, is in every case that the velocity 
tends to vary as the square root of the linear dimensions of the 
organism. 

From all the foregoing discussion we learn that, as Crookes 
once upon a time remarked*, the form as.well as the actions of our 
bodies are entirely conditioned (save for certain exceptions in the 
case of aquatic animals, nicely balanced with the density of the 
surrounding medium) by the strength of gravity upon this globe. 
Were the force of gravity to be doubled, our bipedal form would 
be a failure, and the majority of terrestrial animals would resemble 
short-legged saurians, or else serpents. Birds and insects would 
also suffer, though there would be some compensation for them 
in the increased density of the air. While on the other hand if 
gravity were halved, we should get a lighter, more graceful, more 
active type, requiring less energy and less heat, less heart, less 
lungs, less blood. 

Throughout the whole field of morphology we may find 
examples of a tendency (referable doubtless in each case to some 
definite physical cause) for surface to keep pace with volume, 
through some alteration of its form. The development of “vilh” 
on the inner surface of the stomach and intestine (which enlarge 
its surface much as we enlarge the effective surface of a bath- 
towel), the various valvular folds of the intestinal lining, including. 
the remarkable “spiral fold” of the shark’s gut, the convolutions 
of the brain, whose complexity is evidently correlated (in part 
at least) with the magnitude of the animal,—all these and many 
more are cases in which a more or Jess constant ratio tends to be 
maintained between mass and surface, which ratio would have 
been more and more departed from had it not been for the 
alterations of surface-form 7. 


* Proc. Psychical Soc. xu, pp. 338-355, 1897. 

+ For various calculations of the increase of surface due to histological and 
anatomical subdivision, see E. Babak, Ueber die Oberflachenentwickelung bei 
Organismen, Biol. Centralbl. xxx, pp. 225-239, 257-267, 1910. In connection 
with the physical theory of surface-energy, Wolfgang Ostwald has introduced the 
conception of specific surface, that is to say the ratio of surface to volume, or S/V. 
In a cube, V=/, and S=6/?; therefore S/V=6//. Therefore if the side 7 measure 


1 ieee OF BULK AND SURFACE 33 


In the case of very small animals, and of individual cells, the 
principle becomes especially important, in consequence of the 
molecular forces whose action is strictly limited te the superficial 
layer. In the cases just mentioned, action is facilitated by increase 
of surface: diffusion, for instance, of nutrient liquids or respiratory 
gases is rendered more rapid by the greater area of surface; but 
there are other cases in which the ratio of surface to mass may 
make an essential change in the whole condition of the system. 
We know, for instance, that iron rusts when exposed to moist 
air, but that it rusts ever so much faster, and is soon eaten away, 
if the iron be first reduced to a heap of small filings; this is a 
mere difference of degree. But the spherical surface of the rain- 
drop and the spherical surface of the ocean (though both happen 
to be alike in mathematical form) are two totally different pheno- 
mena, the one due to surface-energy, and the other to that form 
of mass-energy which we ascribe to gravity. The contrast is still 
more clearly seen in the case of waves: for the little ripple, whose 
form and manner of propagation are governed by surface-tension, 
is found to travel with a velocity which is inversely as the square 
root of its length; while the ordinary big waves, controlled by 
gravitation, have a velocity directly proportional to the square 
root of their wave-length. In like manner we shall find that the 
form of all small organisms is largely independent of gravity, and 
largely if not mainly due to the force of surface-tension: either 
as the direct result of the continued action of surface tension on 
the semi-fluid body, or else as the result of its action at a prior 
stage of development, in bringing about a form which subsequent 
chemical changes have rendered rigid and lasting. In either case, 
we shall find a very great tendency in small organisms to assume 
either the spherical form or other simple forms related to ordinary 
inanimate surface-tension phenomena; which forms do not recur 
in the external morphology of large animals, or if they in part 
recur it is for other reasons. 


6cm., the ratio S/V =1, and such a cube may be taken as our standard, or unit 
of specific surface. A human blood-corpuscle has, accordingly, a specific surface 
of somewhere about 14,000 or 15,000. It is found in physical chemistry that 
surface energy becomes an important factor when the specific surface reaches a 
value of 10,000 or thereby. 


Tie (Ele 3 


34 ON MAGNITUDE [CH. 


Now this is a very important matter, and is a notable illustration 
of that principle of similitude which we have already discussed 
in regard to several of its manifestations. We are coming easily 
to a conclusion which will affect the whole course of our argument 
throughout this book, namely that there is an essential difference 
in kind between the phenomena of form in the larger and the 
smaller organisms. I have called this book a study of Growth 
and Form, because in the most familiar illustrations of organic 
form, as in our own bodies for example, these two factors are 
inseparably associated, and because we are here justified in thinking 
of form as the direct resultant and consequence of growth: of 
growth, whose varying rate in one direction or another has pro- 
duced, by its gradual and unequal increments, the successive 
stages of development and the final configuration of the whole 
material structure. But it is by no means true that form and 
growth are in this direct and simple fashion correlative or comple- 
mentary in the case of minute portions of living matter. For in 
the smaller organisms, and in the individual cells of the larger, 
we have reached an order of magnitude in which the intermolecular 
forces strive under favourable conditions with, and at length 
altogether outweigh, the force of gravity, and also those other 
forces leading to movements of convection which are the prevailing 
factors in the larger material aggregate. 

However we shall require to deal more fully with this matter 
in our discussion of the rate of growth, and we may leave it mean- 
while, in order to deal with other matters more or less directly 
concerned with the magnitude of the cell. 

The living cell is a very complex field of energy, and of energy 
of many kinds, surface-energy included. Now the whole surface- 
energy of the cell is by no means restricted to its outer surface; 
for the cell is a very heterogeneous structure, and all its proto- 
plasmic alveoli and other visible (as well as invisible) hetero- 
geneities make up a great system of internal surfaces, at every 
part of which one “phase” comes in contact with another “phase,” 
and surface-energy is accordingly manifested. But still, the 
external surface is a definite portion of the system, with a definite 
“‘phase”’ of its own, and however little we may know of the distri- 
bution of the total energy of the system, it is at least plain that 


II] THE SIZE OF CELLS 35 


the conditions which favour equilibrium will be greatly altered by 
the changed ratio of external surface to mass which a change of 
magnitude, unaccompanied by change of form, produces in the cell. 
In short, however it may be brought aboyt, the phenomenon of 
division of the cell will be precisely what is required to keep 
approximately constant the ratio between surface and mass, and 
to restore the balance between the surface-energy and the other 
energies of the system. When a germ-cell, for instance, divides 
or “segments” into two, it does not increase in mass; at least if 
there be some slight alleged tendency for the egg to increase in 
mass or volume during segmentation, it is very slight indeed, 
generally imperceptible, and wholly denied by some*. The 
development or growth of the egg from a one-celled stage to 
stages of two or many cells, is thus a somewhat peculiar kind 
of growth; it is growth which is limited to increase of surface, 
unaccompanied by growth in volume or in mass. 

_ In the case of a soap-bubble, by the way, if it divide into two 

bubbles, the volume is actually diminished +, while the surface-area 
is greatly increased. This is due to a cause which we shall have 
to study later, namely to the increased pressure due to the greater 
curvature of the smaller bubbles. 

An immediate and-remarkable result of the principles just 
described is a tendency on the part of all cells, according to their 
kind, to vary but little about a certain mean size, and to have, 
in fact, certain absolute limitations of magnitude. 

Sachst pointed out, in 1895, that there is a tendency for each 
nucleus to be only able to gather around itself a certain definite 
amount of protoplasm. Driesch§, a little later, found that, by 
artificial subdivision of the egg, it was possible to rear dwarf 
sea-urchin larvae, one-half, one-quarter, or even one-eighth of their 

* Though the entire egg is not increasing in mass, this is not to say that its 
living protoplasm is not increasing all the while at the expense of the reserve 
material, 

+ Cf. Tait, Proc. R.S.EL. v, 1866, and v1, 1868. 

t Physiolog. Notizen (9), p. 425, 1895. Cf. Strasbiirger, Ueber die Wirkungs- 
sphire der Kerne und die Zellgrésse, Histolog. Beitr. (5), pp. 95-129, 1893; 
J. J. Gerassimow, Ueber die Grésse des Zellkernes, Beith. Bot. Centralbl. xvm, 
1905; also G. Levi and T. Terni, Le variazioni dell’ indice plasmatico-nucleare 


durante 1 intercinesi, Arch. Ital. di Anat. x,.p. 545, 1911. 
§ Arch. f. Entw. Mech. iv, 1898, pp. 75, 247. 


36 ON MAGNITUDE [CH. 


normal size; and that these dwarf bodies were composed of only a 
half, a quarter or an eighth of the normal number of cells. Similar 
observations have been often repeated and amply confirmed. For 
instance, in the development of Crepidula (a little American 
“slipper-limpet,”” now much at home on our own oyster-beds), 
Conklin* has succeeded in rearing dwarf and giant individuals, 
of which the latter may be as much as twenty-five times as big 
as the former. But nevertheless, the individual cells, of skin, gut, 
liver, muscle, and of all the other tissues, are just the same size 
in one as in the other,—in dwarf and in giantt. Driesch has laid 
particular stress upon this principle of a “fixed cell-size.” 

We get an excellent, and more familiar illustration of the same 
principle in comparing the large brain-cells or ganglion-cells, both 
of the lower and of the higher animalst. 

In Fig. 1 we have certain identical nerve-cells taken from 
various mammals, from the mouse to the elephant, all represented 
on the same scale of magnification; and we see at once that they 
are all of much the same order of magnitude. The nerve-cell of 
the elephant is about twice that of the mouse in linear dimensions, 
and therefore about eight times greater in volume, or mass. But 
making some allowance for difference of shape, the near dimen- 
sions of the elephant are to those of the mouse in a ratio certainly 
not less than one to fifty; from which it would follow that the 
bulk of the larger animal is something like 125,000 times that of 
the less. And it also follows, the size of the nerve-cells being 


* Conklin, E. G., Cell-size and nuclear-size, J. Hap. Zool. x1t, pp. 1-98, 1912. 

+ Thus the fibres of the crystalline lens are of the same size in large and small 
dogs; Rabl, Z. f. w. Z. uxvu, 1899. Cf. (int. al.) Pearson, On the Size of the Blood- 
corpuscles in Rana, Biometrika, v1, p. 403, 1909. Dr Thomas Young caught sight 
of the phenomenon, early in last century: “The solid particles of the blood do 
not by any means vary in magnitude in the same ratio with the bulk of the animal,” 
Natural Philosophy, ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales were . 
aware of it a hundred years before. But in this case, though the blood-corpuscles 
show no relation of magnitude to the size of the animal, they do seem to have some 
relation to its activity. At least the corpuscles in the sluggish Amphibia are much 
the largest known to us, while the smallest are found among the deer and other 
agile and speedy mammals. (CE. Gulliver, P.Z.S. 1875, p. 474, etc.) This apparent 
correlation may have its bearing on modern views of the surface-condensation 
or adsorption of oxygen in the blood-corpuscles, a process which would be greatly 
facilitated and intensified by the increase of surface due to their minuteness. 

t Cf. P. Enriques, La forma come funzione della grandezza: Ricerche sui 
gangli nervosi degli Invertebrati, Arch. f. Hntw. Mech. xxv, p. 655, 1907-8. 


It] THE SIZE OF CELLS 37 


about as eight to one, that, in corresponding parts of the nervous 
system of the two animals, there are more than 15,000 times as 
many individual cells in one as in the other. In short we may 
(with Enriques) lay it down as a general law that among animals, 
whether large or small, the ganglion-cells vary in size within 
narrow limits; and that, amidst all the great variety of structural 
type of ganglion observed in different classes of animals, it is 
always found that the smaller species have simpler gangha than 
the larger, that is to say ganglia containing a smaller number 
of cellular elements *. The bearing of such simple facts as this 
upon the cell-theory in general is not to be disregarded; and the 


Elephant Horse Cat Rabbit 
= 
2 © ‘3 ) ©) : a ‘ane 
i, C Rat Mouse 
Man Dog 


Fig. 1. Motor ganglion-cells, from the cervical spinal cord. 
(From Minot, after Irving Hardesty.) 


warning is especially clear against exaggerated attempts to 
correlate physiological processes with the visible mechanism of 
associated cells, rather than with the system of energies, or the 
field of force, which is associated with them. For the life of 


* While the difference in cell-volume is vastly less than that between the 
volumes, and very much less also than that between the surfaces, of the respective 
animals, yet there 7s a certain difference; and this it has been attempted to correlate 
with the need for each cell in the many-celled ganglion of the larger animal to 
possess a more complex “‘exchange-system” of branches, for intercommunication 
with its more numerous neighbours. Another explanation is based on the fact 
that, while such cells as continue to divide throughout life tend to uniformity of 
size in all mammals, those which do not do so, and in particular the ganglion cells, 
continue to grow, and their size becomes, therefore, a function of the duration of 
life. Cf. G. Levi, Studii sulla grandezza delle cellule, Arch. Ital. di Anat. e di 
Embryolog. Vv, p. 291, 1906. 


38 ON MAGNITUDE [CH. 


the body is more than the swm of the properties of the cells of 
which it is composed: as Goethe said, “Das Lebendige ist zwar 
in Elemente zerlegt, aber man kann es aus diesen nicht wieder 
zusammenstellen und beleben.”’ 

Among certain lower and microscopic organisms, such for 
instance as the Rotifera, we are still more palpably struck by the 
small number of cells which go to constitute a usually complex 
organ, such as kidney, stomach, ovary, etc. We can sometimes 
number them in a few units, in place of the thousands that make 
up such an organ in larger, if not always higher, animals. These 
facts constitute one among many arguments which combine to 
teach us that, however important and advantageous the subdivision 
of organisms into cells may be from the constructional, or from 
the dynamical point of view, the phenomenon has less essential 
importance in theoretical biology than was once, and is often still, 
assigned to it. 

Again, just as Sachs shewed that there was a limit to the amount 
of cytoplasm which could gather round a single nucleus, so Boveri 
has demonstrated that the nucleus itself has definite limitations 
of size, and that, in cell-division after fertilisation, each new 
nucleus has the same size as its parent-nucleus*. 

In all these cases, then, there are reasons, partly no doubt 
physiological, but in very large part purely physical, which set 
limits to the normal magnitude of the organism or of the cell. 
But as we have already discussed the existence of absolute and 
definite limitations, of a physical kind, to the possible increase in 
magnitude of an organism, let us now enquire whether there be 
not also a lower limit, below which the very existence of an 
organism is impossible, cr at least where, under changed conditions, 
its very nature must be profoundly modified. 

Among the smallest of known organisms we have, for instance, 
Micromonas mesnili, Bonel, a flagellate infusorian, which measures 
about -34y, or -00034 mm., by -00025mm.; smaller even than 
this we have a pathogenic micrococcus of. the rabbit, M. pro- 
grediens, Schréter, the diameter of which is said to be only -00015 
mm. or ‘15, or 1°5 x 10-°cem.,—about equal to the thickness of 


* Boveri, Zellen-studien, V. Ueber die Abhdngigkeit der Kerngrésse und Zellen- 
zahl der Seeigellarven von der Chromosomenzahl der Ausgangszellen. Jena, 1905. 


1] THE LEAST OF ORGANISMS 39 


the thinnest gold-leaf; and as small if not smaller still are a few 
bacteria and their spores. But here we have reached, or all but 
reached the utmost limits of ordinary microscopic vision; and 
there remain still smaller organisms, the so-called “ filter-passers,”’ 
which the ultra-microscope reveals, but which are mainly brought 
within our ken only by the maladies, such as hydrophobia, foot- 
and-mouth disease, or the “mosaic” disease of the tobacco-plant, 
to which these invisible micro-organisms give rise*. Accordingly, 


A 


Fig. 2. Relative magnitudes of: A, human blood-corpuscle (7-5 in diameter) ; 
B, Bacillus anthracis (4—15 x 1p); C, various Micrococei (diam. 0:5—Iy, 
rarely 24); D, Micromonas progrediens, Schroter (diam. 0-15 ,). 


since it is only by the diseases which they occasion that these 
tiny bodies are made known to us, we might be tempted to 
suppose that innumerable other invisible organisms, smaller and 
yet smaller, exist unseen and unrecognised by man. 

To illustrate some of these small magnitudes I have adapted 
the preceding diagram from one given by Zsigmondyt. Upon the 

* Recent important researches suggest that such ultra-minute “ filter-passers” 
are the true cause of certain acute maladies commonly ascribed to the presence 
of much larger organisms; cf. Hort, Lakin and Benians, The true infective 
Agent in Cerebrospinal Fever, etc., J. Roy. Army Med. Corps, Feb. 1916. 


+ Zur Erkenntniss der Kolloide, 1905, p. 122; where there will be found an 
interesting discussion of various molecular and other minute magnitudes, 


40 ON MAGNITUDE [CH. 


same scale the minute ultramicroscopic particles of colloid gold 
would be represented by the finest dots which we could make 
visible to the naked eye upon the paper. 

A bacillus of ordinary, typical size is, say, |» in length. The 
length (or height) of a man is about a millon and three-quarter 
times as great, i.e. 1-75 metres, or 1-75 x 10®w; and the mass of 
the man is in the neighbourhood of five million, million, million 
(5 x 1048) times greater than that of the bacillus. If we ask 
whether there may not exist organisms as much less than the 
bacillus as the bacillus is less than the dimensions of a man, it 
is very easy to see that this is quite impossible, for we are rapidly 
approaching a point where the question of molecular dimensions, 
and of the ultimate divisibility of matter, begins to call for our 
attention, and to obtrude itself as a crucial factor in the case. 

Clerk Maxwell dealt with this matter in his articie “ Atom*,” 
and, in somewhat greater detail, Errera discusses the question on 
the following lnest. The weight of a hydrogen molecule is, 
according to the physical chemists, somewhere about 8-6 x 2 x 10-* 
milligrammes; and that of any other element, whose molecular 
weight is WM, is given by the equation 

Gi) S-bex x 102 
Accordingly, the weight of the atom of sulphur may be taken as 
8-6 x 32 x 10-2 mgm. = 275 x 10-* mgm. 

The analysis of ordinary bacteria shews them to consist of 
about 85 % of water, and 15 % of solids; while the solid residue 
of vegetable protoplasm contains about one part in a thousand 
of sulphur. We may assume, therefore, that the living protoplasm 
contains about 


woo X HS = 15 x 10-° 
parts of sulphur, taking the total weight as = 1. 
But our little micrococcus, of 0-15 uw in diameter, would, if it 
were spherical, have a volume of 


a O15? nO 18) xl 0 *scubice microns: 


* Encyclopaedia Britannica, 9th edit., vol. 1, p. 42, 1875. Ne 

+ Sur la limite de petitesse des organismes, Bull. Soc. R. des Sc méd. et nat. 
de Bruvelles, Jan. 1903; Rec. d awuvres (Physiol. générale), p. 325. 

t Cf. A. Fischer, Vorlesungen iiber Bakterien, 1897, p. 50. 


11] THE LEAST OF ORGANISMS 41 


and therefore (taking its density as equal to that of water), a 
weight of 
PepelorscdOn? = 1654 10c4 mem: 


But of this total weight, the sulphur represents only 
ees Tae 105P 2 sx NO-t aga. 
And if we divide this by the weight of an atom of sulphur, we have 
27 x 10-17 = 275 x 10-8 = 10,000, or thereby. 


According to this estimate, then, our little Micrococcus progrediens 
should contain only about 10,000 atoms of sulphur, an element 
indispensable to its protoplasmic constitution; and it follows that 
an organism of one-tenth the diameter of our micrococcus would 
only contain 10 sulphur-atoms, and therefore only ten chemical 
“molecules” or units of protoplasm! 

It may be open to doubt whether the presence of sulphur be 
really essential to the constitution of the proteid or “ protoplasinic ~ 
molecule; but Errera gives us yet another illustration of a 
similar kind, which is free from this objection or dubiety. The 
molecule of albumin, as is generally agreed, can scarcely be less 
than a thousand times the size of that of such an element as 
sulphur: according to one particular determination*, serum 
albumin has a constitution corresponding to a molecular weight 
of 10,166, and even this may be far short of the true complexity 
of a typical albuminoid molecule. ‘The weight of such a molecule is 


Ss0ee LONG So 10m 3-7 One mem: 
Now the bacteria contain about 14°, of albuminoids, these 
constituting by far the greater part of the dry residue; and 
therefore (from equation (5)), the weight of albumin in our micro- 
coccus is about 

ae LS x 10-44 2-5) x 10-B mem. 
If we divide this weight by that which we have arrived at as the 
weight of an albumin molecule, we have 

2-5 x 10-3 = 8-7 x 10-18 = 2-9 x 104, 
in other words, our micrococcus apparently contains something 
less than 30,000 molecules of albumin. 


* F. Hofmeister, quoted in Cohnheim’s Chemie der Eiweisskérper, 1900, p. 18. 


42 ON MAGNITUDE [CH. 


According to the most recent estimates, the weight of the 
hydrogen molecule is somewhat less than that on which Errera 
based his calculations, namely about 16 x 10-7? mgms. and 
according to this value, our micrococcus would contain just about 
27,000 albumin molecules. In other words, whichever determina- 
tion we accept, we see that an organism one-tenth as large as our 
micrococcus, in linear dimensions, would only contain some thirty 
molecules of albumin; or, in other words, our micrococcus is only 
about thirty times as large, in linear dimensions, as a single albumin 
molecule *. 

We must doubtless make large allowances for uncertainty in 
the assumptions and estimates upon which these calculations are 
based; and we must also remember that the data with which the 
physicist provides us in regard to molecular magnitudes are, to 
a very great extent, maximal values, above which the molecular 
magnitude (or rather the sphere of the molecule’s range of motion) 
is not likely to he: but below which there is a greater element of 
uncertainty as to its possibly greater minuteness. But nevertheless, 
when we shall have made all reasonable allowances for uncertainty 
upon the physical side, it will still be clear that the smallest known 
bodies which are described as organisms draw nigh towards 
molecular magnitudes, and we must recognise that the subdivision 
of the organism cannot proceed to an indefinite extent, and in all 
probability cannot go very much further than it appears to have 
done in these already discovered forms. For, even.after giving 
all due regard to the complexity of our unit (that is to say the 
albumin-molecule), with all the increased possibilities of inter- 
relation with its neighbours which this complexity implies, we 
cannot but see that physiologically, and comparatively speaking, 
we have come down to a very simple thing. . 

While such considerations as these, based on the chemical 
composition of the organism, teach us that there must be a definite 
lower limit to its magnitude, other considerations of a purely 
physical kind lead us to the same conclusion. For our discussion 
of the principle of similitude has already taught us that, long 
before we reach these almost infinitesimal magnitudes, the 


* McKendrick arrived at a still lower estimate, of about 1250 proteid molecules 
in the minutest organisms. Brii. Ass. Rep. 1901, p. 808. 


11] THE LEAST OF ORGANISMS 43, 


diminishing organism will have greatly changed in all its physical 
relations, and must at length arrive under conditions which must 
surely be incompatible with anything such as we understand by 
life, at least in its full and ordinary development and manifestation. 

We are told, for instance, that the powerful force of surface- 
tension, or capillarity, begins to act within a range of about 
1/500,000 of an inch, or say 0-05. A soap-film, or a film of oil 
upon water, may be attenuated to far less magnitudes than this; 
the black spots upon a soap-bubble are known, by various con- 
cordant methods of measurement, to be only about 6 x 10-7 cm., 
or about -006 » thick, and Lord Rayleigh and M. Devaux* have 
obtained films of oil of -002 4, or even -001 w in thickness. 

But while it is possible for a fluid film to exist in these almost 
molecular dimensions, it is certain that, long before we reach 
them, there must arise new conditions of which we have little 
knowledge and which it is not easy even to imagine. 

It would seem that, in an organism of «1 p in diameter, or even 
rather more, there can be no essential distinction between the 
interior and the surface layers. No hollow vesicle, I take it, can 
exist of these dimensions, or at least, if it be possible for it to do 
so, the contained gas or fluid must be under pressures of a formid- 
able kind}, and of which we have no knowledge or experience. 
Nor, I imagine, can there be any real complexity, or heterogeneity, 
of its fluid or semi-fluid contents; there can be no vacuoles within 
such a cell, nor any layers defined within its fluid substance, for 
something of the nature of a boundary-film is the necessary 
condition of the existence of such layers. Moreover, the whole 
organism, provided that it be fluid or semi-fluid, can only be 
spherical in form. What, then, can we attribute, in the way of 
properties, to an organism of a size as small as, or smaller than, 
say:05? It must, in all probability, be a homogeneous, structure- 
less sphere, composed of a very small number of albuminoid or 
other molecules. Its vital properties and functions must be 
extraordinarily limited ; its specific outward characters, even if we 
could see it, must be nzl; and its specific properties must be little 
more than those of an ion-laden corpuscle, enabling it to perform 


* Cf. Perrin, Les Atomes, 1914, p. 74. 
+ Cf. Tait, On Compression of Air in small Bubbles, Proc. R. S. H. v, 1865. 


tt ON MAGNITUDE Kee 


this or that chemical reaction, or to produce this or that patho- 
genic effect. Even among inorganic, non-living bodies, there 
must be a certain grade of minuteness at which the ordinary 
properties become modified. For instance, while under ordinary 
circumstances crystallisation starts in a solution about a minute 
solid fragment or crystal of the salt, Ostwald has shewn that we 
may have particles so minute that they fail to serve as a nucleus 
for crystallisation,—which is as much as to say that they are too 
minute to have the form and properties of a “crystal”; and again, 
in his thin oil-films, Lord Rayleigh has noted the striking change 
of physical properties which ensues when the film becomes 
attenuated to something less than one close-packed layer of 
molecules *. 

Thus, as Clerk Maxwell put it, “molecular science sets us face 
to face with physiological theories. Jt forbids the physiologist 
from imagining that structural details of infinitely small dimensions 
‘such as Leibniz assumed, one within another, ad infinitum] 
can furnish an explanation of the infinite variety which exists in 
the properties and functions of the most minute organisms.’ 
And for this reason he reprobates, with not undue severity, those 
advocates of pangenesis and similar theories of heredity, who 
would place “a whole world of wonders within a body so small 
and so devoid of visible structure as a germ.” But indeed it 
scarcely needed Maxwell’s criticism to shew forth the immense 
physical difficulties of Darwin’s theory of Pangenesis: which, 
after all, is as old as Democritus, and is no other than that 
Promethean particulam undique desectam of which we have read, 
and at which we have smiled, in our Horace. 

There are many other ways in which, when we “make a long 
excursion into space,” we find our ordinary rules of physical 
behaviour entirety upset. A very familiar case, analysed by 
Stokes, is that the viscosity of the surrounding medium has a 
relatively powerful effect upon bodies below a certain size. 
A droplet of water, a thousandth of an inch (25 w) in diameter, 
cannot fall in still air quicker than about an inch and a half per 
second; and as its size decreases, its resistance varies as the 
diameter, and not (as with larger bodies) as the surface of the 


¢ 


* Phil. Mag. xivut, 1899; Collected Papers, 1v, p. 430. 


11] OF MOLECULAR MAGNITUDES 45 


drop. Thus a drop one-tenth of that size (2-5), the size, 
apparently, of the drops of water in a light cloud, will fall a 
hundred times slower, or say an inch a minute; and one again 
a tenth of this diameter (say -25 yu, or about twice as big, in linear 
dimensions, as our micrococcus), will scarcely fall an inch in two 
hours. By reason of this principle, not only do the smaller 
bacteria fall very slowly through the air, but all minute bodies 
meet with great proportionate resistance to their movements in 
a fluid. Even such comparatively large organisms as the diatoms 
and the foraminifera, laden though they are with a heavy shell 
of flint or lime, seem to be poised in the water of the ocean, and 
fall in it with exceeding slowness. 

The Brownian movement has also to be reckoned with,—that 
remarkable phenomenon studied nearly a century ago (1827) by 
Robert Brown, facile princeps botanicorum. It is one more of those 
fundamental physical phenomena which the biologists have con- 
tributed, or helped to contribute, to the science of physics. 

The quivering motion, accompanied by rotation, and even by 
translation, manifested by the fine granular particles issuing from 
a crushed pollen-grain, and which Robert Brown proved to have 
no vital significance but to be manifested also by all minute 
particles whatsoever, organic and inorganic, was for many years 
unexplained. Nearly fifty years after Brown wrote, it was said 
to be ‘due, either directly to some calorical changes continually 
taking place in the fluid, or to some obscure chemical action 
between the solid particles and the fluid which is indirectly 
promoted by heat*.” Very shortly after these last words were 
written, it was ascribed by Wiener to molecular action, and we 
now know that it is indeed due to the impact or bombardment of 
molecules upon a body so small that these impacts do not for 
the moment, as it were, “average out” to approximate equality 
on all sides. The movement becomes manifest with particles of 
somewhere about 20 in diameter, it is admirably displayed by 
particles of about 12 in diameter, and becomes more marked 
the smaller the particles are. The bombardment causes our 
particles to behave just like molecules of uncommon size, and this 


* Carpenter, The Microscope, edit. 1862, p. 185. 


46 ON MAGNITUDE [CH. 


behaviour is manifested in several ways*. Firstly, we have the 
quivering movement of the particles; secondly, their movement 
backwards and forwards, in short, straight, disjointed paths; 
thirdly, the particles rotate, and do so the more rapidly the smaller 
they are, and by theory, confirmed by observation, it 1s found 
that particles of 1 in diameter rotate on an average through 
100° per second, while particles of 13 ~ in diameter turn through 
only 14° per minute. Lastly, the very curious result appears, that 
in a layer of fluid the particles are not equally distributed, nor do 
they all ever fall, under the influence of gravity, to the bottom. . 
But just as the molecules of the atmosphere are so distributed, 
under the influence of gravity, that the density (and therefore the 
number of molecules per unit volume) falls off in geometrical 
progression as we ascend to higher and higher layers, so is it with 
our particles, even within the narrow limits of the little portion 
of fluid under our microscope. It is only in regard to particles 
of the simplest form that these phenomena have been theoretically 
investigated +, and we may take it as certain that more complex 
particles, such as the twisted body of a Spirillum, would show 
other and still more complicated manifestations. It is at least 
clear that, just as the early microscopists in the days before Robert 
Brown never doubted but that these phenomena were purely 
vital, so we also may still be apt to confuse, in certain cases, the 
one phenomenon with the other. We cannot, indeed, without the 
most careful scrutiny, decide whether the. movements of our 
minutest organisms are intrinsically “vital”’ (in the sense of being 
beyond a physical mechanism, or working model) or not. For ex- 
ample, Schaudinn has suggested that the undulating movements of 
Spirochaete pallida must be due to the presence of a minute, unseen, 
“undulating membrane’; and Doflein says of the same species 
that “sie verharrt oft mit eigenthiimlich zitternden Bewegungen 
zu einem Orte.” Both movements, the trembling or quivering 


* The modern literature on the Brownian Movement is very large, owing to the 
value which the phenomenon is shewn to have in determining the size of the atom. 
For a fuller, but still elementary account, see J. Cox, Beyond the Atom, 1913, 
pp. 118-128; and see, further, Perrin, Les Atomes, pp. 119-189. 

+ Cf. R. Gans, Wie fallen Stabe und Scheiben in einer reibenden Flissigkeit ? 
Miinchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown’sche Bewegung 
nicht kugelférmiger Teilchen, Wiener Ber. 1912, p. 2339. 


Ir], THE BROWNIAN MOVEMENT AT 


movement described by Doflein, and the undulating or rotating 
movement described by Schaudinn, are just such as may be easily 
and naturally interpreted as part and parcel of the Brownian 
phenomenon. 

While the Brownian movement may thus simulate in a deceptive 
way the active movements of an organism, the reverse statement 
also to a certain extent holds good. One sometimes lies awake of 
a summer's morning watching the flies as they dance under the 
ceiling. It is a very remarkable dance. The dancers do not 
whirl or gyrate, either in company or alone; but they advance 
and retire; they seem to jostle and rebound; between the rebounds 
they dart hither or thither in short straight snatches of hurried 
flight; and turn again sharply in a new rebound at the end of each 
little rush. Their motions are wholly “erratic,” independent of 
one another, and devoid of common purpose. This is nothing else 
than a vastly magnified picture, or simulacrum, of the Brownian 
movement; the parallel between the two cases lies in their 
complete irregularity, but this in itself implies a close resemblance. 
One might see the same thing in a crowded market-place, always 
provided that the bustling crowd had no business whatsoever. 
In like manner Lucretius, and Epicurus before him, watched the 
dust-motes quivering in the beam, and saw in them a mimic 
representation, rez simulacrum et imago, of the eternal motions of 
the atoms. Again the same phenomenon may be witnessed under 
the microscope, in a drop of water swarming with Paramoecia or 
suchlike Infusoria ; and here the analogy has been put to a numerical 
test. Following with a pencil the track of each little swimmer, 
and dotting its place every few seconds (to the beat of a metronome), 
Karl Przibram found that the mean successive distances from a 
common base-line obeyed with great .exactitude the “Einstein 
formula,” that is to say the particular form of the “law of chance” 
which is applicable to the case of the Brownian movement*. The 
phenomenon is (of course) merely analogous, and by no means 
identical with the Brownian movement; for the range of motion 
of the little active organisms, whether they be gnats or infusoria, 
is vastly greater than that of the minute particles which are 


* Ueber die ungeordnete Bewegung niederer Thiere, Pfliiger’s Archiv, cit, 
p. 401, 1913. 


48 ON MAGNITUDE [CH. 


passive under bombardment; but nevertheless Przibram is 
inclined to think that even his comparatively large infusoria are 
small enough for the molecular bombardment to be a stimulus, 
though not the actual cause, of their irregular and interrupted 
movements. 

There is yet another very remarkable phenomenon which may 
come into play in the case of the minutest of organisms; and this 
is their relation to the rays of light, as Arrhenius has told us. 
On the waves of a beam of light, a very minute particle (on 
vacuo) should be actually caught up, and carried along with 
an immense velocity; and this “radiant pressure” exercises 
its most powerful influence on bodies which (if they be of 
spherical form) are just about -00016 mm., or -16 ~ in diameter. 
This is just about the size, as we have seen, of some of 
our smallest known protozoa and bacteria, while we have 
some reason to believe that others yet unseen, and perhaps 
the spores of many, are smaller still. Now we have seen that 
such minute particles fall with extreme slowness in air, even at 
ordinary atmospheric pressures: our organism measuring -16 py 
would fall but 83 metres in a year, which is as much as to say 
that its weight offers practically no impediment to its transference, 
by the slightest current, to the very highest regions of the atmo- 
sphere. Beyond the atmosphere, however, it cannot go, until 
some new force enable it to resist the attraction of terrestrial 
gravity, which the viscosity of an atmosphere is no longer at 
hand to oppose. But it is conceivable that our particle may go 
yet farther, and actually break loose from the bonds of earth. 
For in the upper regions of the atmosphere, say fifty miles high, 
it will come in contact with the rays and flashes of the Northern 
Lights, which consist (as Arrhenius maintains) of a fine dust, or 
cloud of vapour-drops, laden with a charge of negative electricity, 
and projected outwards from the sun. As soon as our particle 
acquires a charge of negative electricity it will begin to be repelled 
by the similarly laden auroral particles, and the amount of charge 
necessary to enable a particle of given size (such as our little 
monad of -16 2) to resist the attraction of gravity may be calculated, 
and is found to be such as the actual conditions can easily supply. 
Finally, when once set free from the entanglement of the earth’s 


i] THE PRESSURE OF LIGHT 49 


atmosphere, the particle may be propelled by the “radiant 
pressure” of hight, with a velocity which will carry it.—like 
Uriel gliding on a sunbeam,—as far as the orbit of Mars in 
twenty days, of Jupiter in eighty days, and as far as the nearest 
fixed star in three thousand years! This, and much more, is 
Arrhenius’s contribution towards the acceptance of Lord Kelvin’s 
hypothesis that life may be, and may have been, disseminated 
across the bounds of space, throughout the solar system and the 
whole universe ! 

It may well be that we need attach no great practical importance 
to this bold conception; for even though stellar space be shewn to 
be mare liberum to minute material travellers, we may be sure that 
those which reach a stellar or even a planetary bourne are infinitely, 
or all but infinitely, few. But whether or no, the remote possibilities 
of the case serve to illustrate in a very vivid way the profound 
differences of physical property and potentiality which are 
associated in the scale of magnitude with simple differences of 
degree. 


CHAPTER III 
THE RATE OF GROWTH 


When we study magnitude by itself, apart, that is to say, 
from the gradual changes to which it may be subject, we are 
dealing with a something which may be adequately represented 
by a number, or by means of a line of definite length; it 1s what 
mathematicians call a scalar phenomenon. When we introduce 
the conception of change of magnitude, of magnitude which varies 
as we pass from one direction to another in space, or from one 
instant to another in time, our phenomenon becomes capable of 
representation by means of a line of which we define both the 
length and the direction; it is (in this particular aspect) what is 
called a vector phenomenon. 

When we deal with magnitude in oreliniag to the dimensions 
of space, the vector diagram which we draw plots magnitude in 
one direction against magnitude in another,—length against 
height, for instance, or against breadth; and the result is simply 
what we call a picture or drawing of an object, or (more correctly) 
a “plane projection” of the object. In other words, what we 
call Form is a ratio of magnitudes, referred to direction in space. 

When in dealing with magnitude we refer its variations to 
successive intervals of time (or when, as it is said, we equate it 
with time), we are then dealing with the phenomenon of growth; 
and it is evident, therefore, that this term growth has wide 
meanings. For growth may obviously be positive or negative; 
that is to say, a thing may grow larger or smaller, greater or less; 
and by extension of the primitive concrete signification of the 
word, we easily and legitimately apply it to non-material things, 
such as temperature, and say, for instance, that a body “grows” 
hot or cold. When in a two-dimensional diagram, we represent 
a magnitude (for instance length) in relation to time (or “plot” 


CH. 111] CONCERNING DIMENSIONS Bl 


length against time, as the phrase is), we get that kind of vector 
diagram which is commonly known as a “curve of growth.” We 
perceive, accordingly, that the phenomenon which we are now 


studying is a velocity (whose ‘‘ dimensions”’ are a or 7) ; and 


_this phenomenon we shall speak of, simply, as a rate of growth. 

In various conventional ways we can convert a two-dimensional 
into a three-dimensional diagram. We do so, for example, by 
means of the geometrical method of “perspective” when we 
represent upon a sheet of paper the length, breadth and depth of 
an object in three-dimensional space; but we do it more simply, 
as a rule, by means of “‘contour-lines,” and always when time is 
one of the dimensions to be represented. If we superimpose upon 
one another (or even set side by side) pictures, or plane projections, 
of an organism, drawn at successive intervals of time, we have 
such a three-dimensional diagram, which is a partial] representation 
(limited to two dimensions of space) of the organism’s gradual 
change of form, or course of development; and in such a case 
our contour-lines may, for the purposes of the embryologist, be 
separated by intervals representing a few hours or days, or, for 
the purposes of the palaeontologist, by interspaces of unnumbered 
and innumerable years*. 

Such a diagram represents in two of its three dimensions form, 
and in two, or three, of its dimensions growth; and so we see how 
intimately the two conceptions are correlated or inter-related to 
one another. In short, it is obvious that the form of an animal 
is determined by its specific rate of growth in various directions ; 
accordingly, the phenomenon of rate of growth deserves to be 
studied as a necessary preliminary to the theoretical study of 
form, and, mathematically speaking, organic form itself appears 
to us as a function of time. 


* Sometimes we find one and the same diagram suffice, whether the intervals 
of time be great or small; and we then invoke “Wolff's Law,” and assert that 
the life-history of the individual repeats, or recapitulates, the history of the race. 

+ Our subject is one of Bacon’s “Instances of the Course,” or studies wherein 
we “measure Nature by periods of Time.” In Bacon’s Catalogue of Particular 
Histories, one of the odd hundred histories or investigations which he foreshadowed 
is precisely that which we are engaged on, viz. a ‘‘ History of the Growth and_ Increase 

- of the Body, in the whole and in its parts.” 


4—2 


Or 
bo 


THE RATE OF GROWTH [CH. 


At the same time, we need only consider this part of our 
subject somewhat briefly. Though it has an essential bearing on 
the problems of morphology, it is in greater degree involved with 
physiological problems; and furthermore, the statistical or 
numerical aspect of the question is peculiarly adapted for the 
mathematical study of variation and correlation. On these 
important subjects we shall scarcely touch; for our main purpose 
will be sufficiently served if we consider the characteristics of a 
rate of growth in a few illustrative cases, and recognise that this 
rate of growth is a very important specific property, with its own 
characteristic value in this organism or that, in this or that part 
of each organism, and in this or that phase of its existence. 

The statement which we have just made that “the form of an 
organism is determined by its rate of growth in various directions, * 
is one which calls (as we have partly seen in the foregoing chapter) 
for further explanation and for some measure of qualification. 
Among organic forms we shall have frequent occasion to see that 
form is in many cases due to the immediate or direct action of 
certain molecular forces, of which surface-tension is that which plays 
the greatest part. Now when surface-tension (for instance) causes 
a minute semi-fluid organism to assume a spherical form, or gives 
the form of a catenary or an elastic curve to a film of protoplasm 
in contact with some solid skeletal rod, or when it acts in various 
other ways which are productive of definite contours, this is a pro- 
cess of conformation that, both in appearance and reality, is very 
different from the process by which an ordinary plant or animal 
grows into its specific form. In both cases, change of form is 
brought about by the movement of portions of matter, and in 
both cases it 1s ultimately due to the action of molecular forces ; 
but in the one case the movements of the particles of matter le 
for the most part within molecular range, while in the other we 
have to deal chiefly with the transference of portions of matter 
into the system from without, and from one widely distant part 
of the organism to another. It is to this latter class of phenomena 
that we usually restrict the term growth; and it is in regard to 
them that we are in a position to study the rate of action in 
different directions, and to see that it is merely on a difference 
of velocities that the modification of form essentially depends. 


mt] OF MOLAR AND MOLECULAR FORCES 53 


The difference between the two classes gf phenomena is somewhat 
akin to the difference between the forces which determine the 
form of a rain-drop and those which, by the flowing of the waters 
and the sculpturing of the solid earth, have brought about the 
complex configuration of a river; molecular forces are paramount 
in the conformation of the one, and molar forces are dominant 
in the other. 

At the same time it is perfectly true that all changes of form, 
inasmuch as they necessarily involve changes of actual and relative 
magnitude, may, in a sense, be properly looked upon as phenomena 
of growth; and it is also true, since the movement of matter must 
always involve an element of time*, that in all cases the rate of 
growth is a phenomenon to be considered. Even though the 
molecular forces which play their part in modifying the form of 
an organism exert an action which is, theoretically, all but 
instantaneous, that action is apt to be dragged out to an appreciable 
interval of time by reason of viscosity or some other form of 
resistance in the material. From the physical or physiological 
point of view the rate of action even in such cases may be well 
worth studying; for example, a study of the rate of cell-division 
in a segmenting egg may teach us something about the work done, 
and about the various energies concerned. But in such cases the 
action is, as a rule, so homogeneous, and the form finally attained 
is so definite and so little dependent on the time taken to effect 
it, that the specific rate of change, or rate of growth, does not 
enter into the morphological problem. 

To sum up, we may lay down the following general statements. 
The form of organisms is a phenomenon to be referred in part 
to the direct action of molecular forces, in part to a more complex 
and slower process, indirectly resulting from chemical, osmotic 
and other forces, by which material is introduced into the organism 
and transferred from one part of it to another. It is this latter 
complex phenomenon which we usually speak of as “growth.” 

* Cf. Aristotle, Phys. vi, 5,235 a 11, émel yap amaca ktéyynows év xpdvm, KT. 
Bacon emphasised, in like manner, the fact that “all motion or natural action 
is performed in time: some more quickly, some more slowly, but all in periods 
determined and fixed in the nature of things. Even those actions which seem 


to be performed suddenly, and (as we say) in the twinkling of an eye, are found 
to admit of degree in respect of duration.” Nov. Org. XLVI. 


54 THE RATE OF GROWTH _ fou. 


Every growing organism, and every part of such a growing 
organism, has its own specific rate of growth, referred to a particular 
direction. It is the ratio between the rates of growth in various 
directions by which we must account for the external forms of 
all, save certain very minute, organisms. This ratio between 
rates of growth in various directions may sometimes be of a 
simple kind, as when it results in the mathematically definable 
outline of a shell, or in the smooth curve of the margin of a leaf. 
It may sometimes be a very constant one, in which case the 
organism, while growing in bulk, suffers httle or no perceptible 
change in form; but such equilibrium seldom endures for more 
than a season, and when the ratio tends to alter, then we have 
the phenomenon of morphological ‘development,’ or steady and 
persistent change of form. 

This elementary concept of Form, as determined by varying 
rates of Growth, was clearly apprehended by the mathematical 
mind of Haller,—who had learned his mathematics of the great 
John Bernoulli, as the latter in turn had learned his physiology 
from the writings of Borelli. Indeed it was this very point, the 
apparently unlimited extent to which, in the development of the 
chick, inequalities of growth could and did produce changes of 
form and changes of anatomical “structure,” that led Haller to 
surmise that the process was actually without limits, and that all 
development was but an unfolding, or “evolutio,’ m which no 
part came into being which had not essentially existed before *. 
In short the celebrated doctrine of “‘ preformation”’ implied on the 
one hand a clear recognition of what, throughout the later stages 
of development, growth can do, by hastening the increase in size 
of one part, hindering that of another, changing their relative 
magnitudes and positions, and altermg their forms; while on the 
other hand it betrayed a failure (inevitable in those days) to 
recognise the essential difference between these movements of 
masses and the molecular processes which precede and accompany 

* Of. (e.g.) Blem. Physiol. ed. 1766, vu, p. 114, “Ducimur autem ad evolu- 
tionem potissimum, quando a perfecto animale retrorsum progredimur, et incre- 
mentorum atque mutationum seriem relegimus. Ita inveniemus perfectum illud 
animal fuisse imperfectius, alterius figurae et fabricae, et denique rude et informe: 


et tamen idem semper animal sub iis diversis phasibus fuisse, quae absque ullo 
saltu perpetuos parvosque per gradus cohaereant.” 


111] THE DOCTRINE OF PREFORMATION 55 


them, and which are characteristic of another order of magni- 
tude. : 

By other writers besides Haller the very general, though not 
strictly universal connection between form and rate of growth 
has been clearly recognised. Such a connection is implicit in 
those “proportional diagrams” by which Diirer and some of his 
brother artists were wont to illustrate the successive changes of 
form, or of relative dimensions, which attend the growth of the 
child, to boyhood and to manhood. ‘The same connection was 
recognised, more explicitly, by some of the older embryologists, 
for instance by Pander*, and appears, as a survival of the 
doctrine of preformation, in his study of the development of 
the chick. And long afterwards, the embryological aspect of 
the case was emphasised by His, who pointed out, for instance, 
that the various foldings of the blastoderm, by which the neural 
and amniotic folds were brought into being, were essentially 
and obviously the resultant of unequal rates of growth,—of 
local accelerations or retardations of growth,—in what to begin 
with was an even and uniform layer of embryonic tissue. If 
we imagine a flat sheet of paper, parts of which are caused 
(as by moisture or evaporation) to expand or to contract, the 
plane surface is at once dimpled, or “‘buckled,” or folded, by 
the resultant forces of expansion or contraction: and the various 
distortions to which the plane surface of the “germinal disc” is 
subject, as His shewed once and for all, are precisely analogous. 
An experimental demonstration still more closely comparable to 
the actual case of the blastoderm, is obtained by making an 
“artificial blastoderm,” of little pills or pellets of dough, which 
are caused to grow, with varying velocities, by the addition 
of varying quantities of yeast. Here, as Roux is careful to 
point outt, we observe that it is not only the growth of the 
’ individual cells, but the traction exercised through their mutual 
interconnections, which brings about the foldings and other dis- 
tortions of the entire structure. 


* Beitrage zur Entwickelungsgeschichte des Hiihnchens im Hi, p. 40, 1817. Roux 
ascribes the same views also to Von Baer and to R. H. Lotze (Allg. Physiologie, 
p- 353, 1851). 

+ Roux, Die Entwickelungsmechank, p. 99, 1905. 


56 THE RATE OF GROWTH [CH. 


But this again was clearly present to Haller’s mind, and formed 
an essential part of his embryological doctrine. For he has no 
sooner treated of incrementum, or celeritas incrementi, than he 
proceeds to deal with the contributory and complementary pheno- 
mena of expansion, traction (adtractio)*, and pressure, and the 
more subtle influences which he denominates wis derivationis et 
revulsionist: these latter being the secondary and correlated 
effects on growth in one part, brought about, through such 
changes as are produced (for instance) in the circulation, by the 
srowth of another. . 

Let us admit that, on the physiological side, Haller’s or His’s 
methods of explanation carry us back but a little way; yet even 
this little way is something gained. Nevertheless, I can well 
remember the harsh criticism, and even contempt, which His’s 
doctrine met with, not merely on the ground that it was inadequate, 
but because such an explanation was deemed wholly inappropriate, 
and was utterly disavowed. Hertwig, for instance, asserted that, 
in embryology, when we found one embryonic stage preceding 
another, the existence of the former was, for the embryologist, 
an all-sufficient “causal explanation” of the latter. “We consider 
_(he says), that we are studying and explaining a causal relation 
when we have demonstrated that the gastrula arises by invagina- 
tion of a blastosphere, or the neural canal by the infolding of a 
cell plate so as to constitute a tube§.”’ For Hertwig, therefore, as 


* Op. cit. p. 302, **Magnum hoc naturae instrumentum, etiam in corpore 
animato evolvendo potenter operatur; etc.” 

+ Ibid. p. 306. “Subtiliora ista, et aliquantum hypothesi mista, tamen magnum 
mihi videntur speciem veri habere.” 

{ Cf. His, On the Principles of Animal Morphology, Proc. R. S. EH. xv, 
1888, p. 294: ““My own attempts to introduce some elementary mechanical or 
physiological conceptions into embryology have not generally been agreed to by 
morphologists. -To one it seemed ridiculous to speak of the elasticity of the germinal 
layers; another thought that, by such considerations, we ‘put the cart before 
the horse’: and one more recent author states, that we have better things to do 
in embryology than to discuss tensions of germinal layers and similar questions, 
since all explanations must of necessity be of a phylogenetic nature. This opposition 
to the application of the fundamental principles of science to embryological questions 
would scarcely be intelligible had it not a dogmatic background. No other explana- 
tion of living forms is allowed than heredity, and any which is founded on another 
basis must be rejected....... To think that heredity will build organic beings 
without mechanical means is a piece of unscientific mysticism.” 

§ Hertwig, O., Zeit und Streitfragen der Biologie, u, 1897. 


111] OF PHYSICS AND EMBRYOLOGY 57 


Roux remarks, the task of investigating a physical mechanism in 
embryology,—‘‘der Ziel das Wirken zu erforschen,’—has no 
existence at all. For Balfour also, as for Hertwig, the mechanical 
or physical aspect of organic development had little or no attraction. 
In one notable instance, Balfour himself adduced a physical, or 
quasi-physical, explanation of an organic process, when he referred 
the various modes of segmentation of an ovum, complete or partial, 
equal or unequal and so forth, to the varying amount or the 
varying distribution of food yolk in association with the germinal 
protoplasm of the egg*. But in the main, Balfour, hike all the 
other embryologists of his day, was engrossed by the problems of 
phylogeny, and he expressly defined the aims of comparative 
embryology (as exemplified in his own textbook) as being “two- 
fold: (1) to form a basis for Phylogeny. and (2) to form a basis 
for Organogeny or the origin and evolution of organsy.” 

It has been the great service of Roux and his fellow-workers 
of the school of “Entwickelungsmechanik,” and of many other 
students to whose work we-shall refer, to try, as His triedt, to 
import into embryology, wherever possible, the simpler concepts 
of physics, to introduce along with them the method of experiment, 
and to refuse to be bound by the narrow limitations which such 
teaching as that of Hertwig would of necessity impose on the 
work and the thought and on the whole philosophy of the biologist. 


Before we pass from this general discussion to study some of 
the particular phenomena of growth, let me give a single illustration, 
from Darwin, of a point of view which is in marked contrast to 
Haller’s simple but essentially mathematical conception of Form. 

There is a curious passage in the Origin of Species§, where 
Darwin is discussing the leading facts of embryology, and in 
particular Von Baer’s “law of embryonic resemblance.” Here 
Darwin says “We are so much accustomed to see a difference in 


* Cf. Roux, Gesammelte Abhandlungen, 1, p. 31, 1895. 

+ Treatise on Comparative Embryology, t, p. 4, 1881. 

{ Cf. Fick, Anat. Anzeiger, xxv, p. 190, 1904. 

§ Ist ed. p. 444; 6th ed. p. 390. The student should not fail to consult the 
passage in question; for there is always a risk of misunderstanding or mis- 
interpretation when one attempts to epitomise Darwin’s carefully condensed 
arguments. 


58 THE RATE OF GROWTH [CH. 


structure between the embryo and the adult, that we are tempted — 
to look at this difference as in some necessary manner contingent 
on growth. But there is no reason why, for instance, the wing of 
a bat, or the fin of a porpoise, should not have been sketched out with 
all their parts in proper proportion, as soon as any part became 
visible.” After pointing out with his habitual care various 
exceptions, Darwin proceeds to lay down two general principles, 
viz. “that slight variations generally appear at a not very early 
period of life,’ and secondly, that “at whatever age a variation 
first appears in the parent, it tends to reappear at a corresponding 
age in the offspring.” He then argues that it is with nature as 
with the fancier, who does not care what his pigeons look like 
in the embryo, so long as the full-grown bird possesses the desired 
qualities; and that the process of selection takes place when 
the birds or other animals are nearly grown up,—at least on the 
part of the breeder, and presumably in nature as a general rule. 
The illustration of these principles is set forth as follows: “Let 
us take a group of birds, descended from some ancient form and 
modified through natural selection for different habits. Then, 
from the many successive variations having supervened in the 
several species at a not very early age, and having been inherited 
at a corresponding age, the young will still resemble each other 
much more closely than do the adults,—just as we have seen 
with the breeds of the pigeon.... Whatever influence long-continued 
use or disuse may have had in modifying the limbs or other parts 
of any species, this will chiefly or solely have affected it when 
nearly mature, when it was-compelled to use its full powers to 
gain its own living; and the effects thus produced will have been 
transmitted to the offspring at a corresponding nearly mature 
age. Thus the young will not be modified, or will be modified 
only in a slight degree, through the effects of the increased use or 
disuse of parts.’ This whole argument is remarkable, in more 
ways than we need try to deal with here; but it is especially 
remarkable that Darwin should -begin by casting doubt upon the 
broad fact that a “difference in structure between the embryo 
and the adult” is “in some necessary manner contingent on 
growth”; and that he should see no reason why complicated 
structures of the adult “should not have been. sketched out 


111] A PASSAGE IN DARWIN 59 


with all their parts in proper proportion, as soon as any part 
became visible.” It would seem to me that even the most 
elementary attention to form in its relation to growth would have 
removed most of Darwin’s difficulties in regard to the particular 
phenomena which he is here considering. For these phenomena 
are phenomena of form, and therefore of relative magnitude ; 
and the magnitudes in question are attained by growth, proceeding 
with certain specific velocities, and lasting for certain long periods 
of time. And it is accordingly obvious that in any two related 
individuals (whether specifically identical or not) the differences 
between them must manifest themselves gradually, and be but 
little apparent in the young. It is for the same simple reason 
that animals which are of very different sizes when adult, differ 
less and less in size (as well as in form) as we trace them back- 
wards through the foetal stages. 


Though we study the visible effects of varying rates of growth 
throughout wellnigh all the problems of morphology, it is not very 
often that we can directly measure the velocities concerned. 
But owing to the obvious underlying importance which the 
phenomenon has to the morphologist we must make shift to study 
it where we can, even though our illustrative cases may seem to 
have little immediate bearing on the morphological problem*. 

In a very simple organism, of spherical symmetry, such as the 
single spherical cell of Protococcus or of Orbulina, growth is 
reduced to its simplest terms, and indeed it becomes so simple 
in its outward manifestations that it is no longer of special interest 
to the morphologist. The rate of growth is measured by the rate 
of change in length of a radius, i.e. V = (R’ — R)/T, and from 
this we may calculate, as already indicated, the rate of growth in 
terms of surface and of volume: The growing body remains of 
constant form, owing to the symmetry of the system; because, 
that is to say, on the one hand the pressure exerted by the growing 
protoplasm is exerted equally in all directions, after the manner 
of a hydrostatic pressure, which indeed it actually is: while on 
the other hand, the “skin” or surface layer of the cell is sufficiently 


* “Tn omni rerum naturalium historia utile est mensuras definiri et numeros,” 
Haller, Elem. Physiol. m1, p. 258, 1760. Cf. Hales, Vegetable Staticks, Introduction. 


60 THE RATE OF GROWTH Sage 


homogeneous to exert at every point an approximately uniform 
resistance. Under these conditions then, the rate of growth is 
uniform in all directions, and does not affect the form of the 
organism. 

But in a larger or a more complex organism the study of growth, 
and of the rate of growth, presents us with a variety of problems, 
and the whole phenomenon becomes a factor of great morphological 
importance. . We no longer find that it tends to be uniform in 
all directions, nor have we any right to expect that it should. 
The resistances which it meets with will no longer be uniform. 
In one direction but not in others it will be opposed by the 
important resistance of gravity; and within the growing system 
itself all manner of structural differences will come into play, 
setting up unequal resistances to growth by the varying rigidity 
or viscosity of the material substance in one direction or another. 
At the same time, the actual sources of growth, the chemical and 
osmotic forces which lead to the intussusception of new matter, 
are not uniformly distributed; one tissue or one organ may well 
manifest a tendency to increase while another does not; a series 
of bones, their intervening cartilages, and their surrounding 
muscles, may all be capable of very different rates of increment. 
The differences of form which are the resultants of these differences 
in rate of growth are especially manifested during that, part of 
life when growth itself is rapid: when the organism, as we say, 
is undergoing its development. When growth in general has 
become slow, the relative differences in rate between different 
parts of the organism may still exist, and may be made manifest 
by careful-observation, but in many, or perhaps in most cases, the 
resultant change of form does not strike the eye. Great as are 
the differences between the rates of growth in different parts of 
an organism, the marvel is that the ratios between them are so 
nicely balanced as they actually are, and so capable, accordingly, 
of keeping for long periods of time the form of the growing organism 
all but unchanged. There is the nicest possible balance of forces 
and resistances in every part of the complex body; and when 
this normal equilibrium is disturbed, then we get abnormal 
growth, in the shape of tumours, exostoses, and malformations 
of every kind. 


Stature 


1] QUETELET’S ANTHROPOMETRIE 61 


The rate of growth in Man. 


Man will serve us as well as another organism for our first 
illustrations of rate of growth; and we cannot do better than go 
for our first data concerning him to Quetelet’s Anthropométrie*, an 
epoch-making book for the biologist. For not only is it packed 
with information, some of it still unsurpassed, in regard to human 
growth and form, but it also merits our highest admiration as the 
first great essay in scientific statistics, and the first work in which 
organic variation was discussed from the point of view of the 
mathematical theory of probabilities. 


1700; a a okg 
mm. ‘ aie 
1500 +60 


1300 50 


1100 40 
900 30 
700 120 


410 


4 =e 1. 1 4 4___|_ 1 4 4 4 4 re 4 (0) 
OPERA SS V4 9 10 15 20 
yrs. 


Time 


Fig. 3. Curve of Growth in Man, from birth to 20 yrs (3); from Quetelet’s Belgian 
data. The upper curve of stature from Bowditch’s Boston data. 


If the child be some 20 inches, or say 50 cm. tall at birth, and 
the man some six feet high, or say 180 cm., at twenty, we may 
say that his average rate of growth has been (180 — 50)/20 cm., or 
6-5 centimetres per annum. But we know very well that this is 

* Brussels, 1871. Cf. the same author’s Physique sociale, 1835, and Lettres 
sur la théorie des probabilités, 1846. See also, for the general subject, Boyd, R., 
Tables of weights of the Human Body, ete. Phil. Trans. vol. cit, 1861; Roberts, 


C., Manual of Anthropometry, 1878; Daftner, F., Das Wachsthum des Menschen 
(2nd ed.), 1902, etc. 


Weight 


62 THE RATE OF GROWTH [CH. 


but a very rough preliminary statement, and that the boy grew 
quickly during some, and slowly during other, of his twenty years. 
It becomes necessary therefore to study the phenomenon of growth 
in successive small portions; to study, that is to say, the successive 
lengths, or the successive small differences, or increments, of 
length (or of weight, etc.), attained in successive short increments 
of time. This we do in the first instance in the usual way, by 
the “graphic method” of plotting length against time, and so con- 
structing our “curve of growth.” Our curve of growth, whether 
of weight or length (Fig. 3), has always a certain characteristic 
form, or characteristic curvature. This is our immediate proof of 
the fact that the rate of growth changes as time goes on; for had 
it not been so, had an equal increment of length been added in 
each equal interval of time, our “curve” would have appeared 
as a straight line. Such as itis, it tells us not only that the rate 
of growth tends to alter, but that it alters in a definite and orderly 
way; for, subject to various minor interruptions, due to secondary 
causes, our curves of growth are, on the whole, “smooth” curves. 

The curve of growth for length or stature in man indicates 
a rapid increase at the outset, that is to say during the quick 
growth of babyhood; a long period of slower, but still rapid and 
almost steady growth in early boyhood; as a rule a marked 
quickening soon after the boy is in his teens, when he comes to 
“the growing age”; and finally a gradual arrest of growth as the 
boy “comes to his full height,” and reaches manhood. 

If we carried the curve further, we should see a very curious 
thing. We should see that a man’s full stature endures but for 
a spell; long before fifty* it has begun to abate, by sixty it is 
notably lessened, in extreme old age the old man’s frame is 
shrunken and it is but a memory that “he once was tall.”” We 
have already seen, and here we see again, that growth may have 
a “negative value.” The phenomenon of negative growth in old 
age extends to weight also, and is evidently largely chemical in 
origin: the organism can no longer add new material to its fabric 
fast enough to keep pace with the wastage of time. Our curve 


* Dr Johnson was not far wrong in saying that “‘life declines from thirty-five” ; 
though the Autocrat of the Breakfast-table, like Cicero, declares that “the furnace 
is in full blast for ten years longer.” 


1] . OF MAN’S STATURE 63 


of growth is in fact a diagram of activity, or “‘time-energy”’ 
diagram*. As the organism grows it is absorbing energy beyond 
its daily needs, and accumulating it at a rate depicted in our 


Stature, weight, and span of outstretched arms. 
(After Quetelet, pp. 193, 346.) 
Stature in metres Weight in kgm. Span of % ratio 


SSS (ee SS) arms, of stature 
Male Female %F/M Male Female %F/M _ male to span 


Age 
0 0-500 0-494 98-8 3:2 2-9 90-7 0-496 100°8 
1 0-698 0-690 98-8 9-4 8:8 93-6 0-695 100-4 
2 0-791 0-781 98-7 11-3 10-7 94-7 0-789 100-3 
3 0-864 0-854 98-8 12-4 11-8 95-2 0-863 100-1 
4 0-927 0-915 98-7 14-2 13-0 91-5 0-927 100-0 
5 0-987 0-974 98-7 15-8 14-4 HIST! 0-988 99-9 
6 1-046 1-031 98-5 17-2 16-0 93-0 1-048 99-8 
7 1-104 1-087 98-4 19-1 ALES) 91-6 1-107 SSPVi 
8 1-162 1-142 98-2 20:8 1gG=T 91-3 1-166 99-6 


9 1-218 1-196 98-2 22-6 21-4 94-7 1-224 99-5 
10 1-273 1-249 98-1 24-5 23°5 95-9 1-281 99-4 
11 1-325 1-301 98-2 27-1 25-6 94-5 1-335 99-2 
12 1-375 1-352 98-5 29-8 29-8 100-0 1-388 Ciel 


£13 1-423 1-400 98-4 34-4 32-9 95-6 1-438 98-9 


14 1-469 1-446 98-4 38°8 36-7 94-6 1-489 98-7 
15 1-513 1-488 98-3 43-6 40-4 92-7 1-538 99-4 
16 1-554 1-521 97-8 49-7 43:6 87-7 1-584 98-1 
a7 1-594 1-546 97-0 52:8 47-3 89-6 1-630 Vay) 
18 1-630 1-563 95-9 57:8 49-0 84:8 1-670 97-6 
19 1-655 1-570 94-9 58-0 51-6 89-0 1-705 Oat 
20 1-669 1-574 94-3 60-1 52-3 87-0 1-728 96-6 
25 1-682 1-578 93-8 62-9 53°3 84:7 1-731 97-2 
30 1-686 1-580 93-7 63-7 54:3 85:3 1-766 95:5 
40 1-686 1-580 93-7 63-7 55-2 86-7 1-766 95-5 
50 1-686 1-580 93-7 63-5 56-2 88-4 — = 
60 1-676 1-571 93-7 61-9 54:3 87:7 —— = 
70 1-660 1-556 93-7 59-5 51-5 86-5 —— = 
80 1-636 1-534 93-8 57:8 49-4 85-5 — = 
90 1-610 1-510 93°8 57°8 49-3 85:3 — = 


curve; but the time comes when it accumulates no longer, and at 
last it is constrained to draw upon its dwindling store. But in part, 
the slow decline in stature is an expression of an unequal contest 
between our bodily powers and the unchanging force of gravity, 


* Joly, The Abundance of Life, 1915 (1890), p. 86. 


64 THE RATE OF GROWTH [CH. 


which draws us down when we would fain rise up*. For against 
gravity we fight all our days, in every movement of our limbs, in 
every beat of our hearts; it is the indomitable force that defeats 
us in the end, that lays us on our deathbed, that lowers us to the 
grave ft. 

Side by side with the curve which repiesents growth in length, 
or stature, our diagram shows the curve of weightt. That this 
curve is of a very different shape from the former one, is accounted 
for in the main (though not wholly) by the fact which we have 
already dealt with, that, whatever be the law of increment in a 
linear dimension, the law of increase in volume, and therefore in 
weight, will be that these latter magnitudes tend to vary as 
the cubes of the lmear dimensions. This however does not 
account for the change of direction, or “point of inflection” 
which we observe in the curve of weight at about one or two 
years old, nor for certain other differences between our two curves 
which the scale of our diagram does not yet make clear. These 
differences are due to the fact that the form of the child is altering 
with growth, that other lnear dimensions are varying somewhat 
differently from length or stature, and that consequently the 
growth in bulk or weight is following a more complicated Jaw. 

Our curve of growth, whether for weight or length, is a direct 
picture of velocity, for it represents, as a connected series, the 
successive epochs of time at which successive weights or lengths 
are attained. But, as we have already in part seen, a great part 
of the interest of our curve hes in the fact that we can see from 
it, not only that length (or some other magnitude) is changing, 
but that the rate of change of magnitude, or rate of growth, is 
itself changing. We have, in short, to study the phenomenon of 
acceleration: we have begun by studying a velocity, or rate of 

* « Tou pes, méstre de tout [Le poids, maitre de tout], méstre sénso vergougno, 
Que te tirasso en bas de sa brutalo pougno,” J. H. Fabre, Oubreto prouvengalo, p. 61. 

+ The continuity of the phenomenon of growth, and the natural passage from 
the phase of increase to that of decrease or decay, are admirably discussed by 
Enriques, in “‘ La morte,” Riv. di Scienza, 1907, and in “‘Wachsthum und seine 
analytische Darstellung,” Biol. Centralbl. June, 1909. Haller (Hlem. vu, p. 68) 
recognised decrementum as a phase of growth, not less important (theoretically) 
than incrementum: ‘“‘tristis, sed copiosa, haec est materies.” 


t Cf. (int. al.), Friedenthal, H., Das Wachstum des Kérpergewichtes...in 
verschiedenen Lebensaltern, Zeit. f. allg. Physiol. 1x, pp. 487-514, 1909. 


111] VELOCITY AND ACCELERATION 65 


change of magnitude; we must now study an acceleration, or 
rate of change of velocity. The rate, or velocity, of growth is 
measured by the slope of the curve; where the curve is steep, it 
‘means that growth js rapid, and when growth ceases the curve 
appears as a horizontal line. If we can find a means, then, of 
representing at successive epochs the corresponding slope, or 
steepness, of the curve, we shall have obtained a picture of the 
rate of change of velocity, or the acceleration of growth. The 
measure of the steepness of a curve is given by the tangent to 
the curve, or we may estimate it by taking for equal intervals 
of time (strictly speaking, for each infinitesimal interval of time) 
the actual increment added during that interval of time: and in 
practice this simply amounts to taking the successive differences 
between the values of length (or of weight) for the successive 
ages which we have begun by studying. If we then plot these 
successive differences against time, we obtain a curve each point 
upon which represents a velocity, and the whole curve indicates 
the rate of change of velocity, and we call it an acceleration-curve. 
It contains, in truth, nothing whatsoever that was not implicit 
in our former curve; but it makes clear to our eye, and brings 
within the reach of further investigation, phenomena that were 
hard to see in the other mode of representation. 

The acceleration-curve of height, which we here illustrate, in 
Fig. 4, is very different in form from the curve of growth which 
we have just been looking at; and it happens that, in this case, 
there is a very marked difference between the curve which we 
obtain from Quetelet’s data of growth in height and that which 
we may draw from any other series of observations known to me 
from British, French, American or German writers. It begins (as 
will be seen from our next table) at a very high level, such , 
as it never afterwards attains; and still stands too high, during 
the first three or four years of life, to be represented on the scale 
of the accompanying diagram. From these high velocities it falls 
away, on the whole, until the age when growth itself ceases, and 
when the rate of growth, accordingly, has, for some years together, 
the constant value of nil; but the rate of fall, or rate of change of 
velocity, is subject to several changes or interruptions. During 
the first three or four vears of life the fall is continuous and rapid, 


a Gi. 2 


66 THE RATE OF GROWTH [CH. 


but it is somewhat arrested for a while in childhood, from about 
five years old to eight. According to Quetelet’s data, there is 
another slight interruption in the falling rate between the ages of 
about fourteen and sixteen; but in place of this almost insignificant 
interruption, the English and other statistics indicate a sudden 


70 7 $ 


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mm. \ i 
per|annum Pe 
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Sif Sees i | 
EN t 
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T y NS 
2] e Ll 
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a= ie 
a \o 
e Vv N@. 
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= 20} a 
eral 
* 
10; 
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Fig. 4. Mean annual increments of stature (3), Belgian and American. 


,and very marked acceleration of growth beginning at about 
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this period of acceleration is over, the rate begins to fall again, 
and does so with great rapidity. We do not know how far the 
absence of this striking feature in the Belgian curve is due to the 
imperfections of Quetelet’s data, or whether it is a real and 
significant feature in the small-statured race which he investigated. 

Even apart from these data of Quetelet’s (which seem to ~ 
constitute an extreme case), it is evident that there are very — 


67 


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68 THE RATE OF GROWTH [CH. 


marked differences between different races, as we shall presently 
see there are between the two sexes, in regard to the epochs of 
acceleration of growth, in other words, in the “phase” of the 
curve. 

It is evident that, if we pleased, we might represent the rate 
of change of acceleration on yet another curve, by constructing a 
table of “second differences”; this would bring out certain very 
interesting phenomena, which here however we must not stay to 
discuss. 


Annual Increment of Weight in Man (kgm.). 
(After Quetelet, Anthropométrie, p. 346*.) 


Increment Increment 
Age Male Female Age Male Female 
0-1 5-9 5:6 12-13 4-] oo 
]-2 2-0 2-4 13-14 4-0 3°8 
2-3 1:5 1-4 14-15 4-] 37 
3-4 1-5 1:5 15-16 4-2 Bay 
4-5 1-9 1-4 16-17 4-3 33 
5-6 1-9 1-4 17-18 4-2 3-0 
6-7 ieee) IIE 18-19 aad 2:3 
7-8 Jee 1-2 19-20 ict) Hef 
8-9 1-9 2-0 20-21 Sz) MEH 
9-10 ley 2-1 21-22 1-7 0-5 
10-11 1:8 2-4 22-23 1-6 0-4 
11-12 2-0 3°5 23-24 0-9 — 0-2 
12-13 4-] 3°5 24-25 0-8 — 0:2 


The acceleration-curve for man’s weight (Fig. 5), whether we 
draw it from Quetelet’s data, or from the British, American and 
other statistics of later writers, is on the whole similar to that 
which we deduce from the statistics of these latter writers in 
regard to height or stature; that is to say, it is not a curve which 
continually descends, but it indicates a rate of growth which is 
subject to important fluctuations at certain epochs of life. We see 
that it begins at a high level, and falls continuously and rapidly + 

* The values given in this table are not in precise accord with those of the 
Table on p. 63. The latter represent Quetelet’s results arrived at in 1835; the 
former are the means of his determinations in 1835-40. 

7+ As Haller observed it to do in the chick (Hlem. vu, p. 294): ‘‘Hoe iterum 
incrementum miro ordine ita distribuitur, ut in principio incubationis maximum 
est: inde perpetuo minuatur.” 


111 OF BODILY WEIGHT 69 


during the first two or three years of hfe. After a slight recovery, 
it runs nearly level during boyhood from about five to twelve 
years old; it then rapidly rises, in the “growing period” of the 
early teens, and slowly and steadily falls from about the age of 
sixteen onwards. It does not reach the base-line till the man is 
about seven or eight and twenty, for normal increase of weight 
continues during the years when the man is “filling out,” long 
after growth in height has ceased; but at last, somewhere about 
thirty, the velocity reaches zero, and even falls below it, for then 


Kilos per annum 


O}- foe 
aie Sees 1 | n 1 n nl | StS CS Se ie Oe ee Le ee ee eee 
0 5 10 is 20 years 25 


Fig. 5. Mean annual increments of weight, in man and woman; 
from Quetelet’s data. 


the man usually begins to lose weight a little. The subsequent 
slow changes in this acceleration-curve we need not stop to deal 
with. 
In the same diagram (Fig. 5) I have set forth the acceleration- 
curves in respect of increment of weight for both man and woman, 
according to Quetelet. That growth in boyhood and growth in 
girlhood follow a very different course is a matter of common 
knowledge; but if we simply plot the ordinary curve of growth, 
or velocity-curve, the difference, on the small scale of our diagrams, 


70 THE RATE OF GROWTH lon. 


is not very apparent. It is admirably brought out, however, in 
the acceleration-curves. Here we see that, after infancy, say 
from three years old to eight, the velocity in the girl is steady, 
just as in the boy, but it stands on a lower level in her case than 
in his: the little maid at this age is growing slower than the boy. 
But very soon, and while his acceleration-curve is still represented 
by a straight line, hers has begun to ascend, and until the girl 
is about thirteen or fourteen it continues to ascend rapidly. 
After that age, as after sixteen or seventeen in the boy’s case, it 
begins to descend. In short, throughout all this period, it is a very 
stmilar curve in the two sexes; but it has its notable differences, 
in amplitude and especially in phase. Last of all, we may notice 
that while the acceleration-curve falls to a negative value in the 
male about or even a little before the age of thirty years, this 
does not happen among women. They continue to grow in 
weight, though slowly, till very much later in life; until there 
comes a final period, in both sexes alike, during which weight, 
and height and strength all alike diminish. 

From certain corrected, or “typical” values, given for American children 
by Boas and Wissler (/.c. p. 42), we obtain the following still clearer comparison 
of the annual increments of stature in boys and girls: the typical stature at 
the commencement of the period, i.e. at the age of eleven, being 135-1 cm. 


and 136-9 cm. for the bane and girls respectively, and the annual increments 
being as follows: 


Age 12 13) la ee LOR iS Sal Ome 
Boys (em.) 4-] 6:38 le Oe 1be2 332) 9D 0-One Ors 
Girls (em.) 7-5 7:0 46 2-1 09 04 0-1 0:0 0-0 
Difference -3°4 -0O7 4:1 58 43 2°83 18 09 03 


The result of these differences (which are essentially phase- 
differences) between the two sexes in regard to the velocity of 
growth and to the rate of change of that velocity, is to cause the 
ratio between the weights of the two sexes to fluctuate in a some- 
what complicated manner. At birth the baby-girl weighs on the 
average nearly 10 per cent. less than the boy. Tull about two 
years old she tends to gain upon him, but she then loses again 
until the age of about five; from five she gains for a few years 
somewhat rapidly, and the girl of ten to twelve is only some 
3 per cent. less in weight than the boy. The boy in his teens gains 


rt] OF BODILY WEIGHT 71 


steadily, and the young woman of twenty is nearly 15 per cent. 
lighter than the man. This ratio of difference again slowly 
diminishes, and between fifty and sixty stands at about 12 per 
cent., or not far from the mean for all ages; but once more as 
old age advances, the difference tends, though very slowly, to 
increase (Fig. 6). 

While careful observations on the rate of growth in other 
animals are somewhat scanty, they tend to show so far as they 
go that the general features of the phenomenon are always much 
the same. Whether the animal be long-lived, as man or the 
elephant, or short-lived, like horse or dog, it passes through the 


Bearing Bet ee ee ai ! 
0) 10 20 30 40 50 60 70 80 90 


years 


Fig. 6. Percentage ratio, throughout life, of female weight to male; 
from Quetelet’s data. 


same phases of growth*. In all cases growth begins slowly; it 
attains a maximum velocity early in its course, and afterwards 
slows down (subject to temporary accelerations) towards a point 
where growth ceases altogether. But especially in the cold- 
blooded animals, such as fishes, the slowing-down period is very 
greatly protracted, and the size of the creature would seem never 
actually to reach, but only to approach asymptotically, to a 
maximal limit. 

The size ultimately attained is a resultant of the rate, and of 


* There is a famous passage in Lucretius (Vv, 883) where he compares the course 
of life, or rate of growth, in the horse and his boyish master: Principio circum 
tribus actis impiger annis Floret equus, puer hautquaquam, etc. 


72 THE RATE OF GROWTH [CH. 


the duration, of growth. It is in the main true, as Minot has 
said, that the rabbit is bigger than the guinea-pig because he 
grows the faster; but that man is bigger than the rabbit because 
he goes on growing for a longer time. 


In ordinary physical investigations dealing with velocities, as 
for instance with the course of a projectile, we pass at once from 
the study of acceleration to that of momentum and so to that of 
force; for change of momentum, which is proportional to force, 
is the product of the mass of a body into its acceleration or change 
of velocity. But we can take no such easy road of kinematical 
investigation in this case. The “velocity” of growth is a very 
different thing from the “velocity” of the projectile. The forces 
at work in our case are not susceptible of direct and easy treatment ; 
they are too varied in their nature and too indirect in their action 
for us to be justified in equating them directly with the mass of 
the growing structure. 


It was apparently from a feeling that the velocity of growth ought in some 
way to be equated with the mass of the growing structure that Minot* intro- 
duced a curious, and (as it seems to me) an unhappy method of representing 
erowth, in the form of what he called “‘ percentage-curves ” ; a method which has 
been followed by a number of other writers and experimenters. Minot’s method 
was to deal, not with the actual increments added in successive periods, such 
as years or days, but with these increments represented as percentages of the 
amount which had been reached at the end of the former period. For instance, 
taking Quetelet’s values for the height in centimetres of a male infant from 
birth to four years old, as follows: 


Years 0 1 2 s 4 
cm. 50-0 69-8 79:1 86-4 92- 


Minot would state the percentage growth in each of the four annual periods 
at 39-6, 13-3, 9-6 and 7-3 per cent. respectively. 

Now when we plot actual length against time, we have a perfectly definite 
thing. When we differentiate this L/7’, we have dL/dT’, which is (of course) 
velocity; and from this, by a second differentiation, we obtain d?L/dT?, that 
is to say, the acceleration. 


* Minot, C. 8., Senescence and Rejuvenation, Journ. of Physiol. xu, pp. 97- 
153, 1891; The Problem of Age, Growth and Death, Pop. Science Monthly 
(June—Dec.), 1907. 


a) 


mt] OF PRE-NATAL AND POST-NATAL GROWTH 73 


But when you take percentages of y, you are determining dy/y, and when 
you plot this against dx, you have 


1d 
dyly or dy or dy 


aan y.. dx’ : y da’ 


that is to say, you are multiplying the thing you wish to represent by another 
quantity which is itself continually varying; and the result is that you are 
dealing with something very much less easily grasped by the mind than the 
original factors. Professor Minot is, of course, dealing with a perfectly 
legitimate function of x and y; and his method is practically tantamount to 
plotting log y against x, that is to say, the logarithm of the increment against 
the time. This could only be defended and justified if it led to some simple 
result, for instance if it gave us a straight line, or some other simpler curve 
than our usual curves of growth. As a matter of fact, it is manifest that it 
does nothing of the kind. 


Pre-natal and post-natal growth. 


In the acceleration-curves which we have shown above (Figs. 
2, 3), it will be seen that the curve starts at a considerable interval 
from the actual date of birth; for the first two increments which 
we can as yet compare with one another are those attained during 
the first and second complete years of life. Now we can in many 
cases “interpolate” with safety between known points upon a 
curve, but it is very much less safe, and is not very often justifiable 
(at least until we understand the physical principle involved, and 
its mathematical expression), to “extrapolate” beyond the limits 
of our observations. In short, we do not yet know whether our 
curve continued to ascend as we go backwards to the date of 
birth, or whether it may not have changed its direction, and 
descended, perhaps, to zero-value. In regard to length, or 
stature, however, we can obtain the requisite information from 
certain tables of Riissow’s*, who gives the stature of the infant 
month by month during the first year of its life, as follows: 


Age in months OM mece oe 4a ieee Oe lO Ih 12 
Length in em. (50) 54 58 60 62 64 65 66 67-5 68 69 70-5 72 
| Differences (in cm.) Ae OO) yo tel, Chere fit Wl Tbe 1:5 | 


If we multiply these monthly differences, or mean monthly 
velocities, by 12, to bring them into a form comparable with the 


* Quoted in Vierordt’s Anatomische...Daten und Tabellen, 1906. p. Ss 


74 THE RATE OF GROWTH [CH. 


annual velocities already represented on our acceleration-curves, 
we shall see that the one series of observations joins on very well 
with the other; and in short we see at once that our acceleration- 
curve rises steadily and rapidly as we pass back towards the date 
of birth. 

But birth itself, in the case of a viviparous animal, is but an 
unimportant epoch in the history of growth. It is an epoch whose 
relative date varies according to the particular animal: the foal 


cms, 


Epoch of Birth 


OL 8 eA: abe) On BOM MIaaInIG IS" soOma 


months 


Fig. 7. Curve of growth (in length or stature) of child, before and after 
birth. (From His and Riissow’s data.) 


and the lamb are born relatively later, that is to say when develop- 
ment has advanced much farther, than in the case of man: the 
kitten and the puppy are born earlier and therefore more helpless 
than we are; and the mouse comes into the world still earlier 
and more inchoate, so much so that even the little marsupial is 
scarcely more unformed and embryonic. In all these cases alike, 
we must, in order to study the curve of growth in its entirety, 
take full account of prenatal or intra-uterine growth. 


tit] OF PRE-NATAL AND POST-NATAL GROWTH 75 


According to His*, the following are the mean lengths of the 
unborn human embryo, from month to month. 


Months OF rt DS AE 5 6 7 8 9 10 
(Birth) 
Lengthinmm. 0 75 40 84 162 275 352 402 443 472 490) 
500 J 
Increment per — 75 32°5 44 78 1138 77-50 41 29 = 18) 
month in mm. 28 J 


These data link on very well to those of Riissow, which we 
have just considered, and (though His’s measurements for the 


1 9 T Tecamal a T oat if TT Teemint =f 
cms. 
per|month | 
10 MS 4 
& 
Q 
8 S | 
: 
6 Q 4 


; 7) 
4 6 Bea O a Oe AS 6 Sige 20.22 


months 


Fig. 8. Mean monthly increments of length or stature of child (in cms.). 


pre-natal months are more detailed than are those of Riissow for 
the first year of post-natal life) we may draw a continuous curve of 
growth (Fig. 7) and curve of acceleration of growth (Fig. 8) for the 
combined periods. It will at once be seen that there is a “ point 
of inflection” somewhere about the fifth month of intra-uterine 
life: up to that date growth proceeds with a continually increasing 

* Unsere Kérperform, Leipzig, 1874. 

+ No such point of inflection appears in the curve of weight according to 


C. M. Jackson’s data (On the Prenatal Growth of the Human Body, etc., Amer. 
Journ. of Anat. 1x, 1909, pp. 126 156), nor in those quoted by him from Ahlfeld, 


76 THE RATE OF GROWTH [CH. 


velocity; but after that date, though growth is still rapid, its 
velocity tends to fall away. There is a slight break between our 
two separate sets of statistics at the date of birth, while this is 
the very epoch regarding which we should particularly like to 
have precise and continuous information. Undoubtedly there is 
a certain slight arrest of growth, or diminution of the rate of 
growth, about the epoch of birth: the sudden change in the 


500 
mm. 
400}- 
Length 
300)/- 
= 
= 
ia) 
3S 
200F <= 
o 
3 
wr 
100- 
~~ Acceleration 
oa 
> Eee 
l et 
16) 8 10 
months 


Fig. 9. Curve of pre-natal growth (length or stature) of child; and 
corresponding curve of mean monthly increments (mm.). 


Fehling and others. But it is plain that the very rapid increase of the monthly 
weights, approximately in the ratio of the cubes of the corresponding lengths, 
would tend to conceal any such breach of continuity, unless it happened to be very 
marked indeed. Moreover in the case of Jackson’s data (and probably also in 
the others) the actual age of the embryos was not determined, but was estimated 
from their lengths. The following is Jackson’s estimate of average weights at 
intervals of a lunar month: 
Months 0 ya t24 53) rt: 5 6 7 8 9 10 
Wtin gms. 0 -04 3 36 120 330 600 1000 1500 2200 3200 


1m] OF PRE-NATAL AND POST-NATAL GROWTH 77 


method of nutrition has its inevitable effect; but this slight 
temporary set-back is ew followed by a secondary, and 
temporary, acceleration. 

It is worth our while to draw a separate curve to illustrate on 
a larger scale His’s careful data for the ten months of pre-natal 
life (Fig. 9). We see that this curve of growth is a beautifully 
regular one, and is nearly symmetrical on either side of that. point 
of inflection of which we have already spoken; it is a curve for 
which we might well hope to find a simple mathematical expression. 
The acceleration-curve shown in Fig. 9 together with the pre-natal 


eee ee eee 
OT Anis 8 10 12 14 16 18 20 22 24 26°26 30 


days 
Fig. 10. Curve of growth of bamboo (from Ostwald, after Kraus), 


curve of growth, is not taken directly from His’s recorded data, 
but is derived from the tangents drawn to a smoothed curve, 
corresponding as nearly as possible to the actual curve of growth: 
the rise to a maximal velocity about the fifth month and the 
subsequent gradual fall are now demonstrated even more clearly 
than before. In Fig. 10, which is a curve of growth of the 
bamboo*, we see (so far as it goes) the same essential features, 


* G. Kraus (after Wallich-Martius), Ann. du Jardin bot. de Burtenzorg, x1t, 1, 
1894, p. 210. Cf. W. Ostwald, Zeitliche Eiyenschaften, etc. p. 56. 


78 THE RATE OF GROWTH [CH. 


the slow beginning, the rapid increase of velocity, the point of 
inflection, and the subsequent slow negative acceleration *. , 


Variability and Correlation of Growth. 


The magnitudes and velocities which we are here dealing with 
are, of course, mean values derived from a certain number, some- 
times: a large number, of individual cases. But no statistical 
account of mean values is complete unless we also take account 
of the amount of variability among the individual cases from which’ 
the mean value is drawn. To do this throughout would lead us 
into detailed investigations which le far beyond the scope of this 
elementary book; but we:may very briefly illustrate the nature 
of the process, in connection with the phenomena of growth 
which we have just been studying. 

It was in connection with these phenomena, in the case of 
man, that Quetelet first conceived the statistical study of variation, 
on lines which were afterwards expounded and developed by 
Galton, and which have grown, in the hands of Karl Pearson and 
others, into the modern science of Biometrics. 

When Quetelet tells us, for instance, that the mean stature 
of the ten-year old boy is 1-273 metres, this implies, according to 
the law of error, or law of probabilities, that all the individual 
measurements of ten-year-old boys group themselves in an orderly 
way, that is to say according to a certain definite law, about this 
mean value of 1-273. When these individual measurements are 
grouped and plotted as a curve, so as to show the number of 
individual cases at each individual length, we obtain a characteristic 
curve of error or curve of frequency; and the “spread” of this 
curve is a measure of the amount of variability in this particular 
case. A certain mathematical measure of this “spread,” as 
described in works upon statistics, is called the Index of Variability, 
or Standard Deviation, and is usually denominated by the letter o. 
It is practically equivalent to a determination of the point upon 
the frequency curve where it changes its curvature on either side 
of the mean, and where, from being concave towards the middle 
line, it spreads out to be convex thereto. When we divide this 


* Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des végétaux, 
Bull. Herb. Boissier (2), v, pp. 615, 616, 1905. 


111] ITS VARIABILITY 79 


value by the mean, we get a figure which is independent of 
any particular units, and which is called the Coefficient of Varia- 
bility. (It is usually multiplied by 100, to make it of a more 
convenient amount; and we may then define this coefficient, CU, 
as = o/M x 100.) 

In regard to the growth of man, Pearson has determined this 
coefliicient of variability as follows: in male new-born infants, 
the coefficient in regard to weight is 15-66, and in regard to 
stature, 6:50; in male adults, for weight 10-83, and for stature, 3°66. 
The amount of variability tends, therefore, to decrease with 
growth or age. 

Similar determinations have been elaborated by Bowditch, by 
Boas and Wissler, and by other writers for intermediate ages, 
especially from about five years old to eighteen, so covering a 
great part of the whole period of growth in man*. 


Coefficient of Variability (o/M x 100) in Man, at various ages. 
Age 5 6 sy 8 9 


Stature (Bowditch) see 4-76 4-60 4-42 4-49 4-40 
» (Boas and Wissler) 4-15 4-14 4-22 4:37 4:33 
Weight (Bowditch) aoe 11-56 10-28 11-08 9-92 11-04 
Age ais is ves 10 iM 12 13 14 
Stature (Bowditch) ats 4-55 4-70 4-90 5-47 5:79 
», (Boas and Wissler) 4-36 4-54. 4-73 5-16 5:57 
Weight (Bowditch) ee 11-60 11-76 13-72 13-60 16-80 
Age are ea ae 15 16 17 18 
Stature (Bowditch) Sete 5:57 4-50 4-55 3-69 
» (Boas and Wissler) 5-50 4-69 4:27 3°94 
Weight (Bowditch) ee 15-32 13-28 12-96 10-40 


The result is very curious indeed. We see, from Fig. 11, 
that the curve of variability is very similar to what we have called 
the acceleration-curve (Fig. 4): that is to say, it descends when the 
rate of growth diminishes, and rises very markedly again when, in 
late boyhood, the rate of growth is temporarily accelerated. We 


* Cf. Fr. Boas, Growth of Toronto Children, Rep. of U.S. Comm. of Education, 
1896-7, pp. 1541-1599, 1898; Boas and Clark Wissler, Statistics of Growth, 
Education Rep. 1904, pp. 25-132, 1906; H. P. Bowditch, Rep. Mass. State Board 
of Health, 1877; K. Pearson, On the Magnitude of certain coefficients of Correlation 
in Man, Pr. R. S. uxvi, 1900. 


80 THE RATE OF GROWTH [CH. 


see, in short, that the amount of variability in stature or in weight 
is a function of the rate of growth in these magnitudes, though 
we are not yet in a position to equate the terms precisely, one with 
another. 


If we take not merely the variability of stature or weight at 
a given age, but the variability of the actual successive increments 
in each yearly period, we see that this latter coefficient of variability 
tends to increase steadily, and more and more rapidly, within 


Coefficients of variability 
iN 


5 10 ats 20. 

yrs. 
Fig. 11. Coefficients of variability of stature in Man (3). from Boas 
and Wissler’s data. 


the limits of age for which we have information; and this pheno- 
menon is, in the main, easy of explanation. For a great part of 
the difference, in regard to rate of growth, between one individual 
and another is a difference of phase,—a difference in the epochs 
of acceleration and retardation, and finally in the epoch when 
growth comes to an end. And it follows that the variability of 
rate will be more and more marked, as we approach and reach 
the period when some individuals still continue, and others have 
aiready ceased, to grow. In the following epitomised table, 


1] ITS VARIABILITY 81 


I have taken Boas’s determinations of variability (co) (op. cit. 
. 1548), converted them into the corresponding coefficients of 
variability (o/M x 100), and then smoothed the resulting numbers. 


Coefficients of Variability in Annual. Increment of Stature. 


Age i 8 Dahm lena LOS ae TAs tlio a 
Boys 17:3 15:8 18-6 19:1 21-0 24:7 29:0 36-2 46-1 
Girls 17:1 17-8 19-2 22-7 25-9 29:3 37-0 448 — 


The greater variability of annual increment in the girls, as 
conipared with the boys, is very marked, and is easily explained 
by the more rapid rate at which the girls run through the several 
phases of the phenomenon. 


Just as there is a marked difference in “phase” between the growth- 
curves of the tworsexes, that is to say a difference in the periods when growth 
is rapid or the reverse, so also, within each sex, will there be room for similar, 
but individual phase-differences. Thus we may have children of accelerated 
development, who at a given epoch after birth are both rapidly growing and 
already “‘big for their age”; and others of retarded development who are 
comparatively small and have not reached the period of acceleration which, 
in greater or less degree, will come to them in turn. In other words, there 
must under such circumstances be a strong positive “coefficient of correlation” 
between stature and rate of growth, and also between the rate of growth in 
one year and the next. But it does not by any means follow that a child who 
is precociously big will continue to grow rapidly, and become a man or woman 
of exceptional stature. On the contrary, when in the case of the precocious 
or ‘‘accelerated”’ children growth has begun to slow down, the backward 
ones may still be growing rapidly, and so making up (more or less completely) 
to the others. In other words, the period of high positive correlation between 
stature and increment will tend to be followed by one of negative correlation. 
This interesting and important point, due to Boas and Wissler*, is confirmed 
by the following table :— 


Correlation of Stature and Increment in Boys and Girls. 
(From Boas and Wissler.) 


Age 6 a 8 9 10 11 12 13 14 15 
Stature ) 112-7 115-5 123-2 127-4 133-2 136-8 142-7 147-3 155-9 162-2 
) 111-4 117-7 121-4 127-9 131-8 136-7 144-6 149-7 153-8 157-2 
) Bee 5:3 4-9 5-1 50 4:7 5-9 7:5 6-2 5-2 
) Bey, Bd) Gy | GRY 6-2 7:2 6-5 a4 Bip) 1-7 
) 25 SILI ‘08 25 ‘18 18 48 "29 --42 —-44 
) “44 “14 24 “47 "18 --18 -—-42 -—-39 --63 “11 


Correlation 


( 
( 
Increment ( 
( 
( 
( 


QWBOWow 


* Lc, p. 42, and other papers there quoted. 


82 THE RATE OF GROWTH [CH. 


A minor, but very curious point brought out by the same investigators 
is that, if instead of stature we deal with height in the sitting posture (or, 
practically speaking, with length of trunk or back), then the correlations 
between this height and its annual increment are throughout negative. In 
other words, there would seem to be a general tendency for the long trunks 
to grow slowly throughout the whole period under investigation. It is a 
well-known anatomical fact that tallness is in the main due not to length of 
body but to length of limb. 


The whole phenomenon of variability in regard to magnitude 
and to rate of increment is in the highest degree suggestive: 
inasmuch as it helps further to remind and to impress upon us 
that specific rate of growth is the real physiological factor which 
we want to get at, of which specific magnitude, dimensions and 
form, and all the variations of these, are merely the concrete and 
visible resultant. But the problems of variability, though they 
are intimately related to the general problem of growth, carry us 
very soon beyond our present limitations. 


Rate of growth in other organisms *. 


Just as the human curve of growth has its shght but well- 
marked interruptions, or variations in rate, coinciding with such 
epochs as birth and puberty, so is it with other animals, and this 
phenomenon is particularly striking in the case of animals which 
undergo a regular metamorphosis. 

In the accompanying curve of growth in weight of the mouse 
(Fig. 12), based on W. Ostwald’s observations}, we see a distinct 
slackening of the rate when the mouse is about a fortnight old, 
at which period it opens its eyes and very soon afterwards is 
weaned. At about six weeks old there is «nother well-marked 
retardation of growth, following on a very rapid period, and 
coinciding with the epoch of puberty. 


* See, for an admirable résumé of facts, Wolfgang Ostwald, Ueber die Zeitliche 
Eigenschaften der Entwickelungsvorgdnge (71 pp.), Leipzig, 1908 (Roux’s Vortrdge, 
Heft v): to which work I am much indebted. A long list of observations on the 
growth-rate of various animals is also given by H. Przibram, Hap. Zoologie, 1913, 
pt 1v (Vitalitat), pp. 85-87. 

+ Cf. St Loup, Vitesse de croissance chez les Souris, Bull. Soc. Zool. Fr. Xvi, 
242, 1893; Robertson, Arch. f. Entwickelungsmech. xxv, p. 587, 1908; Donaldson. 
Boas Memorial Volume. New York, 1906. 


1] ITS PERIODIC RETARDATION 83 


Fig. 13 shews the curve of growth of the silkworm*, during its 
whole lary | life, up to the time of its entering the chrysalis stage. 

The silkworm moults four times, at intervals of about a week, 
the first moult being on the sixth or seventh day after hatching. 
A distinct retardation of growth is exhibited on our curve in the 
case of the third and fourth moults; while a similar retardation 
accompanies the first and second moults also, but the scale of 
our diagram does not render it visible. When the worm is about 
seven weeks old, a remarkable process of “ purgation” takes place, 


16[gms. 
14/ a We 


Weis 


10} 


puberty 


weaning 


Cerhoe 1Oerela (20. 254 GO" 65. 40° 2145. 50 
: days 


Fig. 12. Growth in weight of Mouse. (After W. Ostwald.) 


as a preliminary ‘to entering on the pupal, or chrysalis, stage ; 
and the great and sudden loss of weight which accompanies this 
process is the most marked feature of our curve. 

The rate of growth in the tadpole t (Fig. 14) is likewise marked 
by epochs of retardation, and finally by a sudden and drastic 
change. There is a slight diminution in weight immediately after 


* Luciani e Lo Monaco, Arch. Ital. de Biologie, xxvu, p. 340, 1897. 
+ Schaper, Arch. f. Entwickelungsmech. xtv, p. 356, 1902. Cf. Barfurth, Ver- 
suche tiber die Verwandlung der Froschlarven, Arch. f. mikr. Anat. Xx1x, 1887, 


6—2 


84 THE RATE OF GROWTH [CH. 


the little larva frees itself from the egg; there is a retardation of 
growth about ten days later, when the external gills disappear ; 
and finally, the complete metamorphosis, with the loss of the tail, 


ei 
mgms 
4000 
3000 
Wy 
2000 
1000 IY, 
Wy 
Ill 
I U wy 
A! o 
— = ee te i 
5 10 15 20 Obes 30 5) 40 
days 


Fig. 13 Growth in weight of Silkworm. (From Ostwald, after Luciani 
and Lo Monaco.) 


the growth of the legs and the cessation of branchial respiration, 
is accompanied by a loss of weight amounting to wellnigh half 
the weight of the full-grown larva. 


IIT | ITS PERIODIC RETARDATION 85 


While as a general rule, the better the animals be fed the 
quicker they grow and the sooner they metamorphose, Barfiirth 
has pointed out the curious fact that a short spell of starvation, 
just before metamorphosis is due, appears to hasten the change. 


Metamorphosis 


Loss of external 


Escape from e 


0 10 OQOI BOs. AOE 5060, 770%. (B0...90 
days 


Fig. 14. Growth in weight of Tadpole. (From Ostwald, after Schaper.) 


The negative growth, or actual loss of bulk and weight which 
often, and perhaps always, accompanies metamorphosis, is well 
shewn in the case of the eel*. The contrast of size is great between 


* Joh. Schmidt, Contributions to the Life-history of the Eel, Rapports du Conseil 
Intern. pour V exploration de la Mer, vol. v, pp. 137-274, Copenhague, 1906. 


Fig. 15. Development of Eel; from Leptocephalus larvae to young 
Elver. (From Ostwald after Joh. Schmidt.) 


CH. IT} THE RATE OF GROWTH 87 


the flattened, lancet-shaped Leptocephalus larva and the little 
black cylindrical, almost thread-like elver, whose magnitude is 
less than that of the Leptocephalus in every dimension, even, at 
first, in length (Fig. 15). 

From the higher study of the physiology of growth we learn 
that such fluctuations as we have described are but special inter- 
ruptions in a process which is never actually continuous, but is 
perpetually interrupted in a rhythmic manner*. Hofmeister 
shewed, for instance, that the growth of Spirogyra proceeds by 
fits and starts, by periods of activity and rest, which alternate 
with one another at intervals of so many minutes (Fig. 16). And 


99 


97 


95 


93 


91 


89 


87 


85 


990 240 260 
minutes 


20. 40 60 80 100 120 140 160 180 200 


Fig. 16. Growth in length of Spirogyra. (From Ostwald, after Hofmeister.) 


Bose, by very refined methods of experiment, has shewn that 
plant-growth really proceeds by tiny and perfectly rhythmical 
pulsations recurring at regular intervals of a few seconds of time. 
Fig. 17 shews, according to Bose’s observations}, the growth of 
a crocus, under a very high magnification. The stalk grows by 
little jerks, each with an amplitude of about -002 mm., every 


* That the metamorphoses cf an insect are but phases in a process of 
growth, was firstly clearly recognised by Swammerdam, Biblia Naturae, 1737, 
pp. 6, 579 ete. 

+ From Bose, J. C., Plant Response, London, 1906, p. 417. 


88 THE RATE OF GROWTH (CH. 


twenty seconds or so, and after each little increment there is a 
partial recoil. 


7 


myn. 


o 


@) seconds. 80 


Fig. 17. Pulsations of growth in Crocus. in micro-millimetres. 
(After Bose.) 


The rate of growth of various parts or organs*. 


The differences in regard to rate of growth between various 
parts or organs of the body, internal and external, can be amply 
illustrated in the case of man, and also, but chiefly in regard to 
external form, in some few other creatures}. It is obvious that 
there lies herein an endless field for the mathematical study of 
correlation and of variability, but with this aspect of the case we 
cannot deal. 

In the accompanying table, I shew, from some of Vierordt’s 
data, the relative weights, at various ages, compared with the 
weight at birth, of the entire body, of the brain, heart and liver; 


* This phenomenon, of incrementum inequale, as opposed to incrementum in 
universum, was most carefully studied by Haller: “Incrementum inequale multis 
modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus 
fieri, majorem pulmonem, minimum thymum, etc.” (Hlem. vim (2), p. 34). 

7 See (inter alia) Fischel, A., Variabilitaét und Wachsthum des embryonalen 
Korpers, Morphol. Jahrb. xxtv. pp. 369-404, 1896. Oppel, Vergleichung des 
Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten bei Wirbel- 
thieren, Jena, 1891. Faucon, A., Pesées et Mensurations fetales a différents dges 
de la grossesse. (These.) Paris, 1897. Loisel, G., Croissance comparée en poids 
et en longueur des foetus male et femelle dans l’espece humaine, C. R. Soc. de 
Biologie, Paris, 1903. Jackson, C. M., Pre-natal growth of the human body and 
the relative growth of the various organs and parts, Am. J. of Anat. rx, 1909; 
Post-natal growth and variability of the body and of the various organs in the 
albino rat, zbid. xv, 1913. 


111] OF PARTS OR ORGANS 89 


and also the percentage relation which each of these organs bears, 
at the several ages, to the weight of the whole body. 


Weight of Various Organs, compared with the Total Weight of 
the Human Body (male). (After Vierordt, Anatom. Tabellen, 
pp. 38, 39.) 


Percentage weights compared 


Weight Relative weights of with total body-weights 
OE | EO0 
Age inkg. Body Brain Heart Liver Body Brain’ MHeart Liver 
0 3-1 1 1 1 1 100°-.12:29 > 0°76. 4-57 
1 9-0 2:90 . 2-48 1-75 2:50) 100), 10:50 10:46; 3-70 
a LEO 3°55 2-69 2-20 3:02. 100; ~ 9:32 5) 0°47" 73°89 
Opie bord 4:03 2-91 2-75 3-42 100 8-86 0-52 3-88 
4 14-0 4:52 3-49 3°14 4:15 100 9-50 0-53 4:20 


15-9 5:13 3-32 3°43 3°80 100 7-94 0-51 —3:39 
3°57 3°60 4:34 100 7:63 0-48 3:45 
19-7 6:35 3-54 3°95 4:86 100 6-84 047 3:49 
3°62 =“ 4-02 4-59 100 6:38 0-44 3-01 

OF 23-5 7:58 3-74 4-59 4:95 -100 6:06 0-46 2:99 
10 25-2 8-13 3-70 5:41 5:90) 100.) -5:59 0:51 © 3332 
Uh Aa) Bel. ahtay7 5-97 6:14 100 5:04 0:52 3:22 
12 . 29-0 9-35 3:78 (4:13) 621 100 4-88 (0-34) 3-03 
fae 30°L 10:68, 3-90 6-95 731 100 449 050 3-13 
epee tee OT 283-38 9-16 8:39 100 3:47 058 3:20 
BS 40-2 - 13-29% 3-91 8-45 9-22" - 100) 93°62; 048° 321% 
16° 45°9> 14-81 . 3-77 9-76 945 100 3:16 0-51 2-95 
af) 49-7 16-03. 3-70" 10-63 -10-46 100. 2-84 0:51, 2:98 
ise ooo 17-39 13°73 10-33 3-10-65" 100°. 2:64", 0-46 2:80 
Peay oirG 18-58 3:67 ~ 11-42 ae 11-61 100° 2-43) 0:51": 2-86 
20) (59:5 19-19 3:79 912-94. 11-01, 100. 2:43 0-51 . 2-62 


~1 & Ot 
— 
= 
QO 
Qt 
I 
Ts 


(o2) 
bo 
La? 
lor) 
oS 
ie) 
“1 


Pe Ol-2 19-74. 3°74 12:59 11-48. 100, 2°31 0-49 ~, 2-66 
2 62:9 20:29 3:54 13:24 11:82 100 2-14 0-50 2:66 
> 645 20:81 3-66 12-42 10-79 100 .2-16 0-46 2:37 


| 
| 


3:74 13:09 13:04 100 
3-76 12:74 12°84 100 2:16 0:46 2-75 
* From Quetelet. 


Drwrwnw by 


eo) | 
or 
or) 
bo 
bo 
— 
ww 
or) 


From the first portion of the table, it will be seen that none 
of these organs by any means keep pace with the body as a whole 
in regard to growth in weight; in other words, there must be 
‘some other part of the fabric, doubtless the muscles and the bones, 
which increase more rapidly than the average increase of the body. 
Heart and liver both grow nearly at the same rate, and by the 


/ 


90 THE RATE OF GROWTH (on. 


age of twenty-five they have multiplied their weight at birth by 
about thirteen times, while the weight of the entire body has been 
multiplied by about twenty-one; but the weight of the brain has 
meanwhile been multiplied only about three and a quarter times. 
In the next place, we see the very remarkable phenomenon that 
the brain, growing rapidly till the child is about four years old, then 
grows more much slowly till about eight or nine years old, and 
after that time there is scarcely any further perceptible increase. 
These phenomena are diagrammatically illustrated in Fig. 18. 


22 


Multiples of weight at birth 


) 5 70 15 20 years 25 
Fig. 18. Relative growth in weight (in Man) of Brain, Heart, and 
whole Body. 


Many statistics mdicate a decrease of brain-weight during adult life- 
Boas* was inclined to attribute this apparent phenomenon to our statistical 
methods, and to hold that it could ‘‘hardly be explained in any other way 
than by assuming an increased death-rate among men with very large brains, 
at an age of about twenty years.’’ But Raymond Pearl has shewn that there 
is evidence of a steady and very gradual decline in the weight of the brain 
with advancing age, beginning at or before the twentieth year, and con- 
tinuing throughout adult lifey. 

* Lc. p. 1542. ; 
+ Variation and Correlation in Brain-weight, Biometrika, tv, pp. 13-104, 1905. — 


1] OF PARTS OR ORGANS oe 


The second part of the table shews the steadily decreasing 
weights of the organs in question as compared with the body; 
the brain falling from over 12 per cent. at birth to little over 
2 per cent. at five and twenty; the heart from -75 to :46 per 
cent.; and the liver from 4:57 to 2:75 per cent. of the whole 
bodily weight. 

It is plain, then, that there is no simple and direct relation, 
holding good throughout life, between the size of the body as a 
whole and that of the organs we have just discussed; and the 
changing ratio of magnitude is especially marked in the case of 
the brain, which, as we have just seen, constitutes about one-eighth 
of the whole bodily weight at birth, and but one-fiftieth at five 
and twenty. The same change of ratio is observed in other 
animals, in equal or even greater degree. For instance, Max 
Weber* tells us that in the lion, at five weeks, four months, 
eleven months, and lastly when full-grown, the brain-weight 
represents the following fractions of the weight of the whole 
body, viz. 1/18, 1/80, 1/184, and 1/546. And Kellicott has, in 
like manner, shewn that in the dogfish, while some organs (e.g. 
rectal gland, pancreas, etc.) increase steadily and very nearly 
proportionately to the body as a whole, the brain, and some other 
organs also, grow in a diminishing ratio, which is capable of 
representation, approximately, by a logarithmic curvef. 

But if we confine ourselves to the adult, then, as Raymond 
Pearl has shewn in the case of man, the relation of brain-weight 
to age, to stature, or to weight, becomes a comparatively simple 
one, and may be sensibly expressed by a straight line, or simple 
equation. 

Thus, if W be the brain-weight (in grammes), and A be the 
age, or S the stature, of the individual, then (in the case of Swedish 


males) the following simple equations suffice to give the required 
ratios : 
W = 1487-8 — 1-94 A = 915-06 + 2-868. 

* Die Sdugethiere, p. 117. 

+ Amer. J. of Anatomy, vu, pp. 319-353, 1908. Donaldson (Journ. Comp. 
Neur. and Psychol. xvm, pp. 345-392, 1908) also gives a logarithmic formula for 
brain-weight (y) as compared with body-weight (a), which in the case of the white 
rat is y = 554 + -569 log (w— 8-7), and the agreement is very close. But the 
formula 1s admittedly empirica and as Raymond Pearl says (Amer. Nat. 1909, 
p. 303), ‘‘ no ulterior biological significance is to be attached to it.” 


92 THE RATE OF GROWTH [CH. 


These equations are applicable to ages between fifteen and eighty ; 
if we take narrower limits, say between fifteen and fifty, we can get 
a closer agreement by using somewhat altered constants. In the 
two sexes, and in different races, these empirical constants will be 
greatly changed*. Donaldson has further shewn that the correla- 
tion between brain-weight and body-weight is very much closer — 
in the rat than in manf. 


The falling ratio of weight of brain to body with increase of size or age 
finds its parallel in comparative anatomy, in the general law that the larger 
the animal the less is the relative weight of the brain. 


Weight of Weight of 
entire animal brain 

gms. ems. Ratio 
Marmoset ue ae 335 12-5 126 
Spider monkey cae 1845 126 Tens 
Felis minuta ... hats 1234 23°6 1: 56 
F. domestica ... ae 3300 ol 1: 107 
Leopard see Sas 27,700 164 1: 168 
HNGTOM ee wees wee 0 119,500 219 1: 546 
Elephant ec ine 3,048,000 5430 1: 560 
Whale (Globiocephalus) 1,000,000 2511 1: 400 


For much information on this subject, see Dubois, ‘‘ Abhaingigkeit des 
Hirngewichtes von der Kérpergrésse bei den Saugethieren,” Arch. f. Anthropol. 
xxv, 1897. Dubois has attempted, but I think with very doubtful success, 
to equate the weight of the brain with that of the animal. We may do this, 
in a very simple way, by representing the weight of the body as a power of 
that of the brain; thus, in the above table of the weights of brain and body 
in four species of cat, if we call W the weight of the body (in grammes), and 
w the weight of the brain, then if in all four cases we express the ratio by 
W = w", we find that » is almost constant, and differs little from 2-24 in all 
four species: the values being respectively, in the order of the table 2-36, 
2-24, 2:18, and 2-17. But this evidently amounts to no more than an 
empirical rule; for we can easily see that it depends on the particular scale 
which we have used, and that if the weights had been taken, for instance, 
in kilogrammes or in milligrammes, the agreement or coincidence would not 
have occurred ¢. 


* Biometrika, tv, pp. 18-104, 1904. 

+ Donaldson, H. H., A Comparison of the White Rat with Man in respect to 
the Growth of the entire Body, Boas Memorial Vol., New York, 1906, pp. 5-26. 

t Besides many papers quoted by Dubois on the growth and weight of the 
brain, and numerous papers in Biometrika, see also the following: Ziehen, Th., 
Das Gehirn: Massverhdltnisse, in Bardeleben’s Handb. der Anat. des Menschen, 
Iv, pp. 353-386, 1899. Spitzka, E. A., Brain-weight of Animals with special 
reference to the Weight of the Brain in the Macaque Monkey, J. Comp. Neurol. 


1] OF PARTS OR ORGANS 93 
The Length of the Head in Man at various Ages. 
(After Quetelet, p. 207.) 


Men Women 
Age Total height Head Ratio Height Head* Ratio 
m. m m m 


Birth 0-500 0-111 450° 0-494 0-111 4-45 
1 year 0-698 0-154 4-53 0-690 0-154 4-48 
2 years 0-791 0-173 4-57 0-781 0-172 4-54 


ati! 0-864 0-182 4:74 0-854 0-180 4-74 

a O:987 -— 07192" -b-44) . 0-974 - 0-188. 5-18 
‘0 Means 1-273 0-205 6-21 1:249 0-201 -6-21 
eee LHS 0-215. 7-046. 1-488 10-993)" 46790 
20°, 1-669) =) 0227) 7-85 9 TAS $0290) 7205 
3 ee 1-686 0-228 7-39 1-580 0-221 7-15 
AOKE © 1686 0-228 7-39 1580 0-221 7-15 


* A smooth curve, very similar to this, for the growth in “auricular height” 
of the girl’s head, is given by Pearson, in Biometrika, 11, p. 141. 1904. 


As regards external form, very similar differences exist, which 
however we must express in terms not of weight but of length. 
Thus the annexed table shews the changing ratios of the vertical 
length of the head to the entire stature; and while this ratio 
constantly diminishes, it will be seen that the rate of change is 
greatest (or the coefficient of acceleration highest) between the 
ages of about two and five years. 

In one of Quetelet’s tables (swpra, p. 63), he gives measure- 
ments of the total span of the outstretched arms in man, from 
year to year, compared with the vertical stature. The two 
measurements are so nearly identical in actual magnitude that a 
direct comparison by means of curves becomes unsatisfactory ; 
but I have reduced Quetelet’s data to percentages, and it will be 
seen from Fig. 19 that the percentage proportion of span to 
height undergoes a remarkable and steady change from birth to 
the age of twenty years; the man grows more rapidly in stretch 

of-arms than he does in height, and the span which was less than 


xm, pp. 9-17, 1903. Warneke, P., Mitteilung neuer Gehirn und Ké6rperge- 
wichtsbestimmungen bei Saugern, nebst Zusammenstellung der gesammten bisher 
beobachteten absoluten und relativen Gehirngewichte bei den verschiedenen 
Species, J. f. Psychol. u. Neurol. xm, pp. 355-403, 1909. Donaldson, H. H., On 
the regular seasonal Changes in the relative Weight of the Central Nervous System 
ofthe Leopard Frog, Journ. of Morph. xxtt, pp. 663-694, 1911. 


94 THE RATE OF GROWTH [CH. 


the stature at birth by about 1 per cent. exceeds it at the age of 
twenty by about 4 per cent. After the age of twenty, Quetelet’s 
data are few and irregular, but it is clear that the span goes on 
for a long while increasing in proportion to the stature. How 
far the phenomenon is due to actual growth of the arms and 
how far to the increasing breadth of the chest is not yet: 
ascertained. 


101 

D.C. 

100 
99 


98 


97 


96 


5 10 15 20 
yes. 


Fig. 19. Ratio of stature in Man, to span of outstretched arms. 
(From Quetelet’s data.) 


The differences of rate of growth in different parts of the body 
are very simply brought out by the following table, which shews 
the relative growth of certain parts and organs of a young trout, 
at intervals of a few days during the period of most rapid develop- 
ment. It would not be difficult, from a picture of the little 
trout at any one of these stages, to draw its approximate form 
at any other, by the help of the numerical data here set 
forth*. 

* Cf. Jenkinson, Growth, Variability and Correlation in Young Trout, 
Biometrika, vit, pp. 444-455, 1912. | 


ut] OF PARTS OR ORGANS 95 


a 


Trout (Salmo fario): proportionate growth of various organs. 
(From Jenkinson’s data.) 


Days Total Ist - Ventral 2nd Breadth 
old. length Kye Head dorsal fin dorsal ‘Tail-fin of tail 


foreeloo. > 100, 100: 100. “100... 100, . 100 | 100 

63 129:9 129-4 1483 . 1486 1485 108-4 173:8 155-9 

77 154-9 147-3 189-2 (203-6) (193-6) 139-2 257-9 220-4 

92 173-4 179-4 220-0 (193-2) (182-1) 154-5 307-6 272-2 

HOG 104-6 92:5) 242°5> 73:2 — 165:3 173-4 337-3 | 287-7 

While it is inequality of growth in different directions that we 
can most easily comprehend as a phenomenon leading to gradual 
change of outward form, we shall see in another chapter* that 
differences of rate at different parts of a longitudinal system, 
though always in the same direction, also lead to very notable 
and regular transformations. Of this phenomenon, the difference 
in rate of longitudinal growth between head and body is a simple 
case, and the difference which accompanies and results from it in 
the bodily form of the child and the man is easy to see. A like 
phenomenon has been studied in much greater detail in the case 
of plants, by Sachs and certain other botanists, after a method 
in use by Stephen Hales a hundred and fifty years before. 

On the growing root of a bean, ten narrow zones were marked 
off, starting from the apex, each zone a millimetre in breadth. 
After twenty-four hours’ growth, at a certain constant tempera- 
ture, the whole marked portion had grown from 10 mm. to 33 mm. 
in length; but the individual zones had grown at very unequal 
rates, as shewn in the annexed table {. 


Zone Increment Zone Increment 
mm. mm. 
Apex 1-5 6th 13 
2nd 5°8 7th 0-5 
3rd 8-2 8th 0-3 
4th 35 9th 0-2 
5th 1-6 10th 0-1 
Pei chap.) xvi, p. 739. 
y “...[ marked in the same manner as the Vine, young Honeysuckle shoots, 
ete....; and I found in them all a gradual scale of unequal extensions, those parts 


extending most which were tenderest,” Vegetable Staticks, Exp. exxiii. 
t From Sachs, Textbook of Botany, 1882, p. 820. 


Jor THE RATE OF GROWTH [CH. 


The several values in this table le very nearly (as we see by 
Fig. 20) in a smooth curve; in other words a definite law, or 
principle of continuity, connects the rates of growth at successive 
points along the growing axis of the root. Moreover this curve, 
-n its general features, is singularly like those acceleration-curves 
which we have already studied, in which we plotted the rate of 
growth against successive intervals of time, as here we have 
plotted it against successive spatial intervals of an actual growing 


Cla a Ts Piha alae 7 


AO 6 Ora eee 


Zones 


Fig. 20. Rate of growth in successive zones near the tip of the bean-root. 


structure. If we suppose for a moment that the velocities of 
growth had been transverse to the axis, instead of, as in this case, 
longitudinal and parallel with it, it is obvious that these same 
velocities would have given us a leaf-shaped structure, of which 
our curve in Fig. 20 (if drawn to a suitable scale) would represent 
the actual outline on either side of the median axis; or, again, 
if growth had been not confined to one plane but symmetrical 
about the axis, we should have had a sort of turnip-shaped root, 


111] OF PARTS OR ORGANS 97 


having the form of a surface of revolution generated by the same 
curve. This then is a simple and not unimportant illustration of 
the direct and easy passage from velocity to form. 


A kindred problem occurs when, instead of “zones” artificially marked out 
in a stem, we deal with the rates of growth in successive actual “internodes” ; 
and an interesting variation of this problem occurs when we consider, not the 
actual growth of the internodes, but the varying number of leaves which they 
successively produce. Where we have whorls of leaves at each node, as in 
Equisetum and in many water-weeds, then the problem presents itself in a 
simple form, and in one such case, namely in Ceratophyllum, it has been 
carefully investigated by Mr Raymond Pearl*. 

It is found that the mean number of leaves per whorl increases with each 
successive whorl; but that the rate of increment diminishes from whorl! to 
whorl, as we ascend the axis. In other words, the increase in the number of 
leaves per whorl follows a logarithmic ratio; and if y be the mean number of 
leaves per whorl, and x the successional number of the whorl from the root 
or main stem upwards, then 


y= A+ Clog (a - a), 
where A, C, and a are certain specific constants, varying with the part of the 
plant which we happen to be considering. On the main stem, the rate of 
change in the number of leaves per whorl is very slow; when we come to the 
small twigs, or “‘tertiary branches,” it has become rapid, as we see from the 
following abbreviated table: 


Number of leaves per whorl on the tertiary branches of Ceratophyllum. 


Position of whorl] A 1 2 3 4 5 6 
Mean number of leaves 6:55 8:07 9-00 9-20 9-75 10-00 
Increment... we — 1-52 93 ‘20 = (55) (-25) 


We have seen that a slow but definite change of form is a 
common accompaniment of increasing age, and is brought about 
as the simple and natural result of an altered ratio between the 
rates of growth in different dimensions: or rather by the pro- 
gressive change necessarily brought about by the difference in 
their accelerations. There are many cases however in which 
the change is all but imperceptible to ordinary measurement, 
and many others in which some one dimension is easily measured, 
but others are hard to measure with corresponding accuracy. 

* Variation and Differentiation in Ceratophyllum, Carneaie Inst. Publica- 
tions, No. 58, Washington, 1907. 


TG. 


x1 


98 THE RATE OF GROWTH [CH. 


For instance, in any ordinary fish, such as a plaice or a haddock, 
the length is not difficult to measure, but measurements of 
breadth or depth are very much more uncertain. In cases such 
as these, while it remains difficult to define the precise nature of 
the change of form, it is easy to shew that such a change is 
taking place if we make use of that ratio of length to weight 
which we have spoken of in the preceding chapter. Assuming, as 
we may fairly do, that weight is directly proportional to bulk or 
volume, we may express this relation in the form W/L? = k, where 
k is a constant, to be determined for each particular case. (W 
and L are expressed in grammes and centimetres, and it is usual 
to multiply the result by some figure, such as 1000, so as to give 
the constant & a value near to unity.) 


Plaice caught in a certain area, March, 1907. Variation of k (the 
weight-length coefficient) with size. (Data taken from the 
Department of Agriculture and Fisheries’ Plaice-Report, 
vol. 1, p. 107, 1908:) 


Sizeinem. Weightingm. W/L*x10,000 W/L (smoothed) 


23 113 92-8 = 

24 128 92-6 94-3 
25 152 97-3 96-1 
26 173 98-4 DCW) 
27 193 98-1 99-0 
28 221 100-6 100-4 
29 250 102-5 101-2 
30 271 100-4 101-2 
31 300 100-7 100-4 
32 328 100-1 99-8 
33 354 98-5 98-8 
34 384 97-7 98-0 
35 419 97-7 97-6 
36 454 97-3 96-7 
37 492 95-2 96-3 
38 529 96-4 95-6 
39 564 95-1 95-0 
40 614 95:9 95-0 
41 647 93:9 93-8 
42 679 91-6 92-5 
43 732 92-1 92-5 
44 800 93-9 94-0 


45 875 96-0 — 


m1] THE WEIGHT-LENGTH COEFFICIENT 99 


4 


Now while this k may be spoken of as a “constant,” having 
a certain mean value specific to each species of organism, and 
depending on the form of the organism, any change to which it 
may be subject will be a very delicate index of progressive changes 
of form; for we know that our measurements of length are, on 
the average, very accurate, and weighing is a still more delicate 
method of comparison than any linear measurement. 

Thus, in the case of plaice, when we deal with the mean values 
for a large number of specimens, and when we are careful to deal 
only with such as are caught in a particular locality and at a par- 
ticular time. we see that k is by no means constant, but steadily 
increases to a maximum, and afterwards slowly declines with the 


90 


Ee 


TS RO SaaS TLs0. AIL e45cme. 


93 25 27 29 


Fig. 21. Changes in the weight-length ratio of Plaice, with increasing size. 


increasing size of the fish (Fig. 21). To begin with, therefore, the 
weight is increasing more rapidly than the cube of the length, and 
it follows that the length itself is increasing less rapidly than some 
other linear dimension; while in later life this condition is reversed. 
The maximum is reached when the length of the fish is somewhere 
near to 30 cm., and it is tempting to suppose that with this “ point 
of inflection” there is associated some well-marked epoch in the 
fish’s life. As a matter of fact, the size of 30 cm. is approximately 
that at which sexual maturity may be said to begin, or is at least 
‘near enough to suggest a close connection between the two 
phenomena. The first step towards further investigation of the 


7—2 


100 THE RATE OF GROWTH [CH. 


apparent coincidence would be to determine the coefficient & of 
the two sexes separately, and to discover whether or not the point 
of inflection is reached (or sexual maturity is reached) at a smaller 
size in the male than in the female plaice; but the material for 
this investigation is at present scanty. 

A still more curious and more unexpected result appears when 
we compare the values of / for the same fish at different seasons of 
the year*. When for simplicity’s sake (as in the accompanying 
table and Fig. 22) we restrict ourselves to fish of one particular 


% 
& 
= 
dS 
® 
Oo 
= 
< 
> 
8 
Q 
a) 


U6 Fy MRAM! ds Oe GATS i AO)o sen Nae 


Fig. 22. Periodic annual change in the weight-length ratio of Plaice. | 


size, it is not necessary to determine the value of k, because a 
change in the ratio of length to weight is obvious enough; but 
when we have small numbers, and various sizes, to deal with, 
the determination of k may help us very much. It will be seen, 
then, that in the case of plaice the ratio of weight to length 
exhibits a regular periodic variation with the course of the seasons. 


* Cf. Lammel, Ueber periodische Variationen in Organismen, Biol. Centralbl. 
XXII, pp. 368-376, 1903. 


coeg| THE WEIGHT-LENGTH COEFFICIENT 101 


Relation of Weight to Length in Plaice of 55 cm. long, from Month 
to Month. (Data taken from the Department of Agriculture 
and Fisheries’ Plaice-Report, vol. 11, p. 92, 1909.) 


Average weight 


in grammes W/L? x 100 W/L? (smoothed) 
Jan. 2039 1-226 1-157 
Feb. 1735 1-043 1-080 
March 1616 0-971 0-989 
April 1585 0-953 0-967 
May 1624 0-976 0-985 
June 1707 1-026 1-005 
July 1686 1-013 1-037 
August 1783 1-072 j 1-042 
Sept. Wise, 1:042° 1-111 
Oct: 2029 1-220 1-160 
Nov. 2026 1-218 1-213 
Dee. 1998 1-201 1:215 


With unchanging length, the weight and therefore the bulk of the 
fish falls off from about November to March or April, and again 
between May or June and November the bulk and weight are 
gradually restored. The explanation is simple, and depends 
wholly on the process of spawning, and on the subsequent building 
up again of the tissues and the reproductive organs. It follows 
that, by this method, without ever seeing a fish spawn, and without 
ever dissecting one to see the state of its reproductive system, we 
can ascertain its spawning season, and determine the beginning 
and end thereof, with great accuracy. 


As a final illustration of the rate of growth, and of unequal 
growth in various directions, I give the following table of data 
regarding the ox, extending over the first three years, or nearly 
so, of the animal’s life. The observed data are (1) the weight of 
the animal; month by month, (2) the length of the back, from the 
occiput to the root of the tail, and (3) the height to the withers. 
To these data I have added (1) the ratio of length to height, 
(2) the coefficient (k) expressing the ratio of weight to the cube of 
the length, and (3) a similar coefficient (k’) for the height of the 
animal. It will be seen that, while all these ratios tend to alter 
continuously, shewing that the animal’s form is steadily altering 
as it approaches maturity, the ratio between length and weight 


102 THE RATE OF GROWTH lon. 


changes comparatively little. The simple ratio between length 
and height increases considerably, as indeed we should expect; 
for we know that in all Ungulate animals the legs are remarkably 


‘Relations between the Weight and certain Linear Dimensions 
of the Ox. (Data from Przbram, after Cornevin*.) 


Agein W,wt. L, length k=W/t ko Se 

months inkg. of back 4H, height L/H x 10 x 10 
0 37 -78 -70 1-114 779 1-079 
1 55:3 94. ‘77 1-221 - -665 1-210 
2 86-3. 1-09 85 1-282 -666 1-406 
SEIT | As207 -94 1-284 -690 1-460 
4 150-3 1-314 -95 1-383 . -662 1-754 
5 (1793 1-404 1:040 1-350 649 1-600 
6 2103 1484 ‘1-087 4-365 644 1-638 
Toes Oa) Moar BOON ke desis -699 1:751 
S 12° 967-3,, 58 1147. 1-378 ‘677 1-791 
9) 282-8: = 1691 1:162 1-395 -664 1-802 
100) 03:7 1-651 1:192 1-385 ‘675 1-793 
LS 387-7 co E6Sa) Egg AO FE- See -674 1-794 
12.4 35097 9-740 (1-288 =) 1-405, -666 1-849 
1S a aae7 1:765 1:254 1-407 -682 1-900 
14 BONS) SCTRA. hs U-2645 0) ae -688 1-938 
15 4059 1-804 1:270 1-420 -692 1-982 
16° 417-9 1-814 1-280. 1-417 -700 2-092 
17 4) A938 -Oi 5 i- 8325 Th UIEDOO I ha a6 -689 1-974 
18 423-9 1859 1-297 1-483 -660 1-943 
19) 2 407-98) 12875 20 B07. de4aes 649 1-916 
D0 i) ASTO ) RBk4 OST 1-437 ‘655 1-944 
D1) 447-9. 7) 1899. SBT 1-433 -661 1-943. 
22 4644 1-901 1:333  —-:1-426 -676 1-960 
23 480'9.*: 1-909 °- 1-845" * 1-419 691 1:977 
24 5009 1-914 1-352 1-416 -714 2-027 
25 5209 1:919 1-359 1-412 737 2-075 
26 = 5341 1:924 1-361 1-414 -750 2-119 
97. bAT-3 1-920 1-368 1-415 762 2-162 
28 5545 1929 1-368 1-415 772 "2-190 
99. 15617 1-920°) 6 1:863< aea5 -782 2-218 
30 . 586-2 1-949 1-383 1-409 -792 2-216 
31 610-7 1:969 1-403 1-408 -800 2-211 
32 625:7 . 1-983 1-490 1-396 -803 2-186 
33. 640-7 1-997 1-437 1-390 “805 2-159 
34 655:7 . 2-011 1:454 = 1-388 -806 2-133 


* Cornevin, Ch., Etudes sur la croissance, Arch. de Physiol. norm. et pathol. 
(5), Iv, p. 477, 1892. 


111] THE WEIGHT-LENGTH COEFFICIENT 103 


long *t birth in comparison with other dimensions of the body. 
It is somewhat curious, however, that this ratio seems to fall off 
a little in the third year of growth, the animal continuing to grow 
in height to a marked degree after growth in length has become 
very slow. The ratio between height and weight is by much the 
most variable of our three ratios; the coefficient W/H® steadily 
increases, and is more than twice as great at three years old as 
it was at birth. This illustrates the important, but obvious fact, 
that the coefficient & is most variable in the case of that 
dimension which grows most uniformly, that is to say most nearly 
in proportion to the general bulk of the animal. In short, the 
successive values of k, as determined (at successive epochs) for 
one dimension, are a measure of the variability of the others. 


From the whole of the foregoing discussion we see that a certain 
definite rate of growth is a characteristic or specific phenomenon, 
deep-seated in the physiology of the organism; and that a very 
large part of the specific morphology of the organism depends upon 
the fact that there is not only an average, or aggregate, rate of 
growth common to the whole, but also a variation of rate in 
different parts of the organism, tending towards a specific rate 
characteristic of each different part or organ. The smallest change 
in the relative magnitudes of these partial or localised velocities 
of growth will be soon manifested in more and more striking 
differences of form. This is as much as to say that the time- 
element, which is implicit in the idea of growth, can never (or 
very seldom) be wholly neglected in our consideration of form*. 
It is scarcely necessary to enlarge here upon our statement, for 
‘not only is the truth of it self-evident, but it will find illustration 
again and again throughout this book. Nevertheless, let us go 
out of our way for a moment to consider it in reference to a 
particular case, and to enquire whether it helps to remove any of 
the difficulties which that case appears to present. 


* Herein lies the easy answer to a contention frequently raised by Bergson, 
and to which he ascribes great importance, that ‘“‘a mere variation of size is one 
thing, and a change of form is another.” Thus he considers ‘“‘a change in the 
form of leaves” to constitute “‘a profound morphological difference.” Creative 
Evolution, p. 71. 


104 THE RATE OF GROWTH [CH. 


In a very well-known paper, Bateson shewed that, among a 
large number of earwigs, collected in a particular locality, the 
males fell into two groups, characterised by large or by small 
tail-forceps, with very few instances of intermediate magnitude. 
This distribution into two groups, according to magnitude, is 
illustrated in the accompanying diagram (Fig. 23); and the 
phenomenon was described, and has been often quoted, as one 
of dimorphism, or discontinuous variation. In this diagram the 
time-element does not appear; but it is certain, and evident, that 
it lies close behind. Suppose we take some organism which is 


150 


ae 


50) 


Number of individuals 


0 5 mm.10 
Length of tail-forceps, in mm. 


Fig. 23. Variability of length of tail-forceps in a sample of Earwigs. 
(After Bateson, P. Z. S. 1892, p. 588.) 


born not at all times of the year (as man is) but at some one 
particular season (for instance a fish), then any random sample 
will consist of individuals whose ages, and therefore whose magni- 
tudes, will form a discontinuous series; and by plotting these 
magnitudes on a curve in relation to the number of individuals 
of each particular magnitude, we obtain a curve such as that 
shewn in Fig. 24, the first practical use of which is to enable us 
to analyse our sample into its constituent ““age-groups,” or in 
other words to determine approximately the age, or ages of the 
fish. And if, instead of measuring the whole length of our fish, 
we had confined ourselves to particular parts, such as head, or 


111] A CASE OF DIMORPHISM 105 


tail or fin, we should have obtained discontinuous curves of 
distribution, precisely analogous to those for the entire animal. 
Now we know that the differences with which Bateson was dealing 
were entirely a question of magnitude, and we cannot help seeing 
that the discontinuous distributions of magnitude represented by 
his earwigs’ tails are just such as are illustrated by the magnitudes 
of the older and younger fish; we may indeed go so far as to say 
that the curves are precisely comparable, for in both cases we see 
a characteristic feature of detail, namely that the “spread” of the 
curve is greater in the second wave than in the first, that is to 


200] 
wn 
= weer 
= 
= 
de 
ro 
‘=| 
= 1}e8) 
(o) 
~ 
o 
ao} 
5 
A 50 

1 1 — 
Ocm. 15 20 25 30 35 40 


Length of fish, in cm. 


Fig. 24. Variability of length of body in a sample of Plaice. 


say (in the case of the fish) in the older as well as larger series. 
Over the reason for this phenomenon, which is simple and all but 
obvious, we need not pause. 

It is evident, then, that in this case of “dimorphism,” the tails 
of the one group of earwigs (which Bateson calls the “high males’’) 
have either grown faster, or have been growing for a longer period 
of time, than those of the “low males.” If we could be certain 
that the whole random sample of earwigs were of one and the 
same age, then we should have to refer the phenomenon of di- 
morphism to a physiological phenomenon, simple in kind (however 
remarkable and unexpected); viz. that there were two alternative 


106 THE RATE OF GROWTH [CH. 


values, very different from one another, for the mean velocity of 
growth, and that the individual earwigs varied around one or 
other of these mean values, in each case according to the law of 
probabilities. But on the other hand, it we could believe that 
the two groups of earwigs were of different ages, then the pheno- 
menon would be simplicity itself, and there would be no more to 
be said about it*. 


Before we pass from the subject of the relative rate of growth 
of different parts or organs, we may take brief note of the fact 
that various experiments have been made to determine whether 
the normal ratios are maintained under altered circumstances of 
nutrition, and especially in the case of partial starvation. For 
instance, it has been found possible to keep young rats alive for 
many weeks on a diet such as is just sufficient to maintain life 
without permitting any increase of weight. The rat of three 
weeks old weighs about 25 gms., and under a normal diet should 
weigh at ten weeks old about 150 gms., in the male, or 115 gms. 
in the female; but the underfed rat is still kept at ten weeks old 
to the weight of 25 gms. Under normal diet the proportions of 
the body change very considerably between the ages of three and 
ten weeks. For instance the tail gets relatively longer; and even 
when the total growth of the rat is prevented by underfeeding, 
the form continues to alter so that this increasing length of the 
tail is still manifest 7. 


* IT do not say that the assumption that these two groups of earwigs were of 
different ages is altogether an easy one; for of course, even in an insect whose 
metamorphosis is so simple as the earwig’s, consisting only in the acquisition of 
wings or wing-cases, we usually take it for granted that growth proceeds no more 
after the final stage. or “adult form” is attained, and further that this adult form 
is attained at an approximately constant age, and constant magnitude. But even 
if we are not permitted to think that the earwig may have grown, or moulted, 
after once the elytra were produced, it seems to me far from impossible, and far 
from unlikely, that prior to the appearance of the elytra one more stage of growth, 
or one more moult took place in some cases than in others: for the number of 
moults is known to be variable in many species of Orthoptera. Unfortunately 
Bateson tells us nothing about the sizes or total lengths of his earwigs; but his 
figures suggest that it was bigger earwigs that had the longer tails; and that the 
rate of growth of the tails had had a certain definite ratio to that of the bodies, 
but not necessarily a simple ratio of equality. 

+ Jackson, C. M., J. of Exp. Zool. xx, 1915, p. 99; cf. also Hans Aron, Unters. 


111] THE EFFECT OF TEMPERATURE 107 


Full-fed Rats. 
Agein Length of Length of Total 


weeks body (mm.) tail (m.) length % of tail 
0 48-7 16-9 65-6 25-8 
1 64:5 29-4 93-9 31:3 
3 90-4 59-1 149-5 39°5 
6 128-0 110-0 5 2380 46-2 
10 173-0 150-0 323-0 46-4 
Underfed Rats. 
6 98-0 712-3 170-3 42-5 
10 99-6 83-9 183-5 45-7 


Again as physiologists have long been aware, there is a marked 
difference in the variation of weight of the different organs, 
according to whether the animal’s total weight remain constant, 
or be caused to diminish by actual starvation; and further striking 
differences appear when the diet is not only scanty, but ill-balanced. 
But these phenomena of abnormal growth, however interesting 
from the physiological view, are of little practical importance to 
the morphologist. 


The effect of temperature*. 


The rates of growth which we have hitherto dealt with are 
based on special investigations, conducted under particular local 
conditions. For instance, Quetelet’s data, so far as we have used 
them to illustrate the rate of growth in man, are drawn from his 
study of the population of Belgium. But apart from that 
“fortuitous” individual variation which we have already con- 
sidered, it is obvious that the normal rate of growth will be found 
to vary, in man and in other animals, just as the average stature 

varies, in different localities, and in different “races.” This 
phenomenon is a very complex one, and is doubtless a resultant 
of many undefined contributory causes; but we at least gain 
something in regard to it, when we discover that the rate of growth 
is directly affected by temperature, and probably by other physical 
liber die Beeinfliissung der Wachstum durch die Ernahrung, Berl. klin. Wochenbl. 
LI, pp. 972-977, 1913, ete. 

* The temperature limitations of life, and to some extent of growth, are summar- 


ised for a large number of species by Davenport, Haxper. Morphology, cc. viii, xviii, 
and by Hans Przibram, Hap. Zoologie, Iv, c. v. 


7 


108 THE RATE OF GROWTH [CH. 


conditions. Réaumur was the first to shew, and the observation 
was repeated by Borinet*, that the rate of growth or development 
of the chick was dependent on temperature, being retarded at _ 
temperatures below and somewhat accelerated at temperatures 
above the normal temperature of incubation, that is to say the 
temperature of the sitting hen. In the case of plants the fact 
that growth is greatly affected by temperature is a matter of 
familiar knowledge; the subject was first carefully studied by 
Alphonse De Candolle, and his results and those of his followers 
are discussed in the textbooks of Botany fF. 


That variation of temperature constitutes only one factor in determining 
the rate of growth is admirably illustrated in the case of the Bamboo. It has 
been stated (by Lock) that in Ceylon the rate of growth of the Bamboo is 
directly proportional to the humidity of the atmosphere: and again (by 
Shibata) that in Japan it is directly proportional to the temperature. The 
two statements have been ingeniously and_ satisfactorily reconciled by 
Blackmant, who suggests that in Ceylon the temperature-conditions are 
all that can be desired, but moisture is apt to be deficient: while in Japan 
there is rain in abundance but the average temperature is somewhat too low. 
‘So that in. the one country it is the one factor, and in the other country it is 
the other, which is essentially variable. 


The annexed diagram (Fig. 25), shewing the growth in length 
of the roots of some common plants during an identical period 
of forty-eight hours, at temperatures varying from about 14° to 
37° C., 1s a sufficient illustration of the phenomenon. We see that 
in all cases there is a certain optimum temperature at which the 
rate of growth is a maximum, and we can also see that on either 
side of this optimum temperature the acceleration of growth, 
positive or negative, with increase of temperature is rapid, while 
at a distance from the optimum it is very slow. From the 
data given by Sachs and others, we see further that this optimum 
temperature is very much the same for all the common plants of 
our own climate which have as yet been studied; in them it is 


* Réeaumur: Lart de faire éclore et élever en toute saison des oiseaux domestiques, 
foit par le moyen de la chaleur du fumier, Paris, 1749. 

+ Cf. (int. al.) de Vries, H., Matériaux pour la connaissance de influence de 
la température sur les plantes, Arch. Néerl. v, 385-401, 1870. Koppen, Warme 
und Pflanzenwachstum, Bull. Soc. Imp. Nat. Moscou, xu, pp. 41-110, 1870. 

t Blackman, F. F.. Ann. of Botany, x1x, p. 281, 1905. 


IIT] THE EFFECT OF TEMPERATURE 109 


somewhere about 26° C. (or say 77° F.), or about the temperature 
of a warm summer's day; while it is found, very naturally, to be 
considerably higher in the case of plants such as the melon or the 
maize, which are at home in warmer regions that our own. 


In a large number of physical phenomena, and in a very marked 
degree in all chemical reactions, it is found that rate of action is 
affected, and for the most part accelerated, by rise of temperature ; 


80 
mm. 
70} Pans ih 
7 if / 
On 3 / 

60 fi gee 

| | Zea eh ; 

I 1 /Cucumis 

L / Melo 
50 v 
40 


30 


Inerement in 48 hours 


20 


10 


14°16 18 20 29 24 26 28 30 32 34 36 3840 


Temp. 


Fig. 25. Relation of rate of growth to temperature in certain plants. 
(From Sachs’s data.) 


and this effect of temperature tends to follow a definite ‘ex- 
ponential” law, which holds good within a considerable range of 
temperature, but is altered or departed from when we pass beyond 
certain normal limits. The law, as laid down by van’t Hoff for 
chemical reactions, is, that for an interval of n degrees the velocity 
varies as 2”, x being called the “temperature coefficient” * for the 
reaction 1n question. 

* For various instances of a “temperature coefficient” in physiological pro- 
cesses, see Kanitz, Zeitschr. f. Elektrochemie, 1907, p. 707; Biol. Centralbl. xxvu1, 


p. ll, 1907; Hertzog, R. O., Temperatureinfluss auf die Entwicklungsgesch- 
windigkeit der Organismen, Zeztschr. f. Elektrochemie, x1, p. 820, 1905; Krogh, 


110 - THE RATE OF GROWTH [CH. 


Van’t Hoftf’s law, which has become a fundamental principle 
of chemical mechanics, is likewise applicable (with certain qualifica- 
tions) to the phenomena of vital chemistry; and it follows that, 
on very much the same lines, we may speak of the “temperature 
coefficient” of growth. At the same time we must remember 
that there is a very important difference (though we can scarcely 
call it a fundamental one) between the purely physical and the 
physiological phenomenon, in that in the former we study (or 
seek and profess to study) one thing at a time, while in the latter 
we have always to do with various factors which intersect and 
interfere; increase in the one case (or change of any kind) tends 
to be continuous, in the other case it tends to be brought to arrest. 
This is the simple meaning of that Law of Optimum, laid down by 
Errera and by Sachs as a general principle of physiology: namely 
that every physiological process which varies (like growth itself) 
with the amount or intensity of some external influence, does so 
according to a law in which progressive increase is followed by 
progressive decrease; in other words the function has its optimum 
condition, and its curve shews a definite maximum. In the case 
of temperature, as Jost puts it, it has on the one hand its accelerat- 
ing effect which tends to follow van’t Hoff’s law. But it has also 
another and a cumulative effect upon the organism: “Sie schadigt 
oder sie ermiidet ihn, und je hoher sie steigt, desto rascher macht 
sie die Schadigung geltend und desto schneller schreitet sie voran.”’ 
It would seem to be this double effect of temperature in the case 
of the organism which gives us our “optimum” curves, which are 
the expression, accordingly, not of a primary phenomenon, but 
of a more or less complex resultant. Moreover, as Blackman and 
others have pointed out, our “optimum” temperature is very 
ill-defined until we take account also of the duration of our experi- 
ment; for obviously, a high temperature may lead to a short, 
but exhausting, spell of rapid growth, while the slower rate 
manifested at a lower temperature may be the best in the end. 
Quantitative Relation between Temperature and Standard Metabolism, Int. 
Zeitschr. f. physik.-chem. Biologie, 1, p. 491, 1914; Piitter, A., Ueber Temperatur- 
koefficienten, Zeitschr. f. allgem. Physiol. xv1, p. 574, 1914. Also Cohen, 
Physical Chemistry for Physicians and Biologists (English edition), 1903; Pike, 


F. H., and Scott, E. L., The Regulation of the Physico-chemical Condition of the 
Organism, American Naturalist, Jan. 1915, and various papers quoted therein. 


111] _ THE EFFECT OF TEMPERATURE Ai 


The mile and the hundred yards are won by different runners; 
and maximum rate of working, and maximum amount of work 
done, are two very different things*. 


In the case of maize, a certain series of experiments shewed that 
the growth in length of the roots varied with the temperature as 
followst : 


Temperature Growth in 48 hours 

SC: mm. 

18-0 1-1 
23°5 10:8 
26-6 29-6 
28:5 26°5 
30-2 64-6 
33°5 69-5 
36-5 20-7 


Let us write our formula in the form 
V teem se 

V, 
Then choosing two values out of the above experimental series 


(say the second and the second-last), we have t = 23-5, n = 10, 
and V, V’ = 10-8 and 69-5 respectively. 


n 


69-5 
e ey eer 
Accordingly 10:8 6-4 = giv. 
Therefore aes or -0806 = log x. 
And, « = 1-204 (for an interval of 1°C.). 


This first approximation might be considerably improved by 
taking account of all the experimental values, two only of which 
we have as yet made use of; but even as it is, we see by Fig. 26 
that it 1s in very fair accordance with the actual results of 
observation, within those particular linuts of temperature to which 
the experiment is confined. 


* Cf. Errera, L., L’Optimum, 1896 (Rec. d’ Oeuvres, Physiol. générale, pp. 338-368, 
1910); Sachs, Physiologie d. Pflanzen, 1882, p. 233; Pfeffer, Pflanzenphysiologie, 
ii, p. 78, 1904; and cf. Jost, Ueber die Reactionsgeschwindigkeit im Organismus, 
Biol. Centralbl. xxvi, pp. 225-244, 1906. 

+ After Képpen, Bull. Soc. Nat. Moscou, xu, pp. 41-110, 1871. 


112 THE RATE OF GROWTH [CH. 


For an experiment on Lupinus albus, quoted by Asa Gray%, 
I have worked out the corresponding coefficient, but a little more 
carefully. Its value I find to be 1-16, or very nearly identical 
with that we have just found for the maize; and the correspondence 
between the calculated curve and the actual observations is now 
a close one. 


0 eS 

ae : 
60 
50 
AO 


30 


20 


Increment in 48 hours 


18 90 D9 D4. 96 98  SONn sou aan 
Temperature 


Fig. 26. Relation of rate of growth to temperature in Maize. Observed 
values (after K6ppen), and calculated curve. 


Since the above paragraphs were written, new data have come to hand. 
Miss I. Leitch has made careful observations of the rate of growth of rootlets 
of the Pea; and I have attempted a further analysis of her principal results. 
In Fig. 27 are shewn the mean rates of growth (based on about a hundred 
experiments) at some thirty-four different temperatures between 0-8° and 
29-3°, each experiment lasting rather less than twenty-four hours. Working 
out the mean temperature coefficient for a great many combinations of these 
values, I obtain a value of 1-092 per C.°, or 2-41 for an interval of 10°, and 
a mean value for the whole series showing a rate of growth of just about 
1 mm. per hour at a temperature of 20°. My curve in Fig. 27 is drawn from 
these determinations; and it will be seen that, while it is by no means exact 
at the lower temperatures, and will of course fail us altogether at very high 


* Botany, p. 387. 

+ Leitch, I., Some Experiments on the Influence of Temperature on the Rate 
of Growth in Pisum sativum, Ann. of Botany, Xxx, pp. 25-46, 1916. (Cf. especially 
Table III, p. 45.) 


11] THE EFFECT OF TEMPERATURE 113 


temperatures, yet it serves as a very satisfactory guide to the relations between 
rate and temperature within the ordinary limits of healthy growth. Miss 
Leitch holds that the curve is not a van’t Hoff curve; and this, in strict accuracy, 
we need not dispute. But the phenomenon seems to me to be one into which 
the van’t Hoff ratio enters largely, though doubtless combined with other 
factors which we cannot at present determine or eliminate. 


ea ea |S ee 
ote Seer 
PREECE HEHE 


20 


12 


o 


(eo) 
Growth per hour, in mm. 


Growth in scale-divisions per half-hour. 


0. 4. & 12° 16 90° 94° 99° 32°9 
Temperature 


Fig. 27. Relation of rate of growth to temperature in rootlets of 
Pea. (From Miss I. Leitch’s data.) 


While the above results conform fairly well to the law of the 
temperature coefficient, it is evident that the imbibition of water 
plays so large a part in the process of elongation of the root or 
stem that the phenomenon is rather a physical than a chemical 
one: and on this account, as Blackman has remarked, the data 
commonly given for the rate of growth in plants are apt to be 


ays (els 8 


114 THE RATE OF GROWTH [oH. 


irregular, and sometimes (we might even say) misleading*. The 
fact also, which we have already learned, that the elongation of a 
shoot tends to proceed by jerks, rather than smoothly, is another 
indication that the phenomenon is not purely and simply a 
chemical one. We have abundant illustrations, however, among 
animals, in which we may study the temperature coefficient under 
circumstances where, though the phenomenon is always compli- 
cated by osmotic factors, true metabolic growth or chemical 
combination plays a larger role. Thus Mlle. Maltaux and Professor 
Massart + have studied the rate of division in a certain flagellate, 
Chilomonas paramoecium, and found the process to take 29 minutes 
at 15° C., 12 at 25°, and only 5 minutes at 35°C. These velocities 
are in the ratio of | : 2-4: 5-76, which ratio corresponds precisely 
to a temperature coefficient of 2-4 for each rise of 10°, or about 
1-092 for each degree centigrade. 

By means of this principle we may throw light on the apparently 
complicated results of many experiments. For instance, Fig. 28 
is an illustration, which has been often copied, of O. Hertwig’s 
work on the effect of temperature on the rate of development of 
the tadpole. 

From inspection of this diagram, we see that the time taken 
to attain certain stages of development (denoted by the numbers 
ITI-VIT) was as follows, at 20° and at 10° C., respectively. 


At 20° At 10° 
Stage III 2-0 6-5 days 
ae Bhp 2-7 SA; 
RN 3-0 LOT hs 
PAE 4:0 edie 5 
= Vi 5-0 LG-S) 
Total 16-7 55-6 


That is to say, the time taken to produce a given result at 


* Blackman, F. F., Presidential Address in Botany, Brit. Ass. Dublin, 1908. 

+ Rec. de VInst. Bot. de Bruxelles, v1, 1906. 

{t Hertwig, O., Einfluss der Temperatur auf die Entwicklung von Rana fusca 
und R. esculenta, Arch. f. mikrosk. Anat. U1, p. 319, 1898. Cf. also Bialaszewicz, 
K., Beitrige z. Kenntniss d. Wachsthumsvorginge bei Amphibienembryonen, 
Bull. Acad. Sci. de Cracovie, p. 783, 1908; Abstr. in Arch. f. Entwicklungsmech. 
xxvin, p. 160, 1909. 


11] THE EFFECT OF TEMPERATURE 115 


10° was (on the average) somewhere about 55-6/16-7, or 3-33, 
times as long as was required at 20°. 


ww 

29) 

28 

if P| 

i eae | 

= EIS 

ee ea | 

Bice Denso | aes 

_ Ra eae Oe 

eee | eS eae ae 

JCS Se 

Jc R SSSR RRR See oe ee 

Pees RSs NAT Stel Ls Tee ia a 

SRO SRGRMRBES oon 

Pe f 

i Ce ee 

| | 

aa aaa eas GS 

jeee Pech a/c 
ne eae 
iy OAR 2 
Bae eae ey AA 
UES BRER SaaS ie! 

é ee VAR aS ae) 

2 Ghipat average 

\sP=a7eLeneer aleae 

leacaberean en 

: ee eee EE ROnEee 

Cezacaeeeseaas i aeeee 

nae Deere ca 
eae Temperature Cenligrade 

2¢° 23° 22° 21° 20° 19° 18° 17° 16° 15° 14° 13° 12° H1° 10° F 8° 7° 6° 5° 4° 3° 2° 4° 


Fig. 28. Diagram shewing time taken (in days), at various temperatures (°C.), 
to reach certain stages of development in the Frog: viz. I, gastrula; II, 
medullary plate; III, closure of medullary folds; IV, tail-bud; V, tail and 
gills; VI, tail-fin; VII, operculum beginning; VIII, do. closing; IX, first 
appearance of hind-legs. (From Jenkinson, after O. Hertwig, 1898.) 


We may then put our equation again in the simple form, 
ih 9 919 


8—2 


116 THE RATE OF GROWTH [CH. 


Or, 10 log « = log 3-33 = -52244. 
Therefore log 7 = -05224, 
and e = 1-128. 


That is to say, between the intervals of 10° and 20°C., if it 
take m days, at a certain given temperature, for a certain stage 
of development to be attained, it will take m x 1-128” days, 
when the temperature is » degrees less, for the same stage to 
be arrived at. 


30 
days 


25 


Time 


of pees |. i a =) 
255 CM sO. 1iDe 10° 5° O 
Temperature 


Fig. 29. Calculated values, corresponding to preceding figure. 


Fig. 29 is calculated throughout from this value; and it will 
be seen that it is extremely concordant with the original diagram, 
as regards all the stages of development and the whole range of 
temperatures shewn: in spite of the fact that the coefficient on 
which it is based was derived by an easy method from a very few 
points in the original curves. 


111] THE EFFECT OF TEMPERATURE EUG 


Karl Peter*, experimenting chiefly on echinoderm eggs, and 
also making use of Hertwig’s experiments on young tadpoles, 
gives the normal temperature coefficients for intervals of 10° C. 
(commonly written Q,9) as follows. 


Sphaerechinus _... 2 Ae 2-15, 
Kehinus’... ie se ee 2-135, 
Rana + a es oe 2°86. 


These values are not only concordant, but are evidently of the 
same order of magnitude as the temperature-coefficient in ordinary 
chemical reactions. Peter has also discovered the very interesting 
fact that the temperature-coefficient alters with age, usually but 
not always becoming smaller as age increases. 


Sphaerechinus; Segmentation @!° = 2-29, 
Later stages me 0s 
Kchinus ; Segmentation Ae Dea) 
Later stages a 3208: 
Rana; Segmentation i 223, 
Later stages » oo. 


Furthermore, the temperature coefficient varies with the 
temperature, diminishing as the temperature rises,—a rule which 
van't Hoff has shewn to hold in ordinary chemical operations. 
Thus, in Rana the temperature coefficient at low temperatures 
may be as high as 5-6: which is just another way of saying that 
at low temperatures development is exceptionally retarded. 


In certain fish, such as plaice and haddock, I and others have 
found clear evidence that the ascending curve of growth is subject 
to seasonal interruptions, the rate during the winter months 
being always slower than in the months of summer: it is as though 
we superimposed a periodic, annual, sine-curve upon the continuous 
curve of growth. And further, as growth itself grows less and less 
from year to year, so will the difference between the winter and 
the summer rate also grow less and less. The fluctuation in rate 


* Der Grad der Beschleunigung tierischer Entwickelung durch erhdhte 
Temperatur, A. f. Hntw. Mech. xx, p. 130, 1905. More recently, Bialaszewicz 
has determined the coefficient for the rate of segmentation in Rana as being 
2-4 per 10°C. 


118 THE RATE OF GROWTH [CH. 


will represent a vibration which is gradually dying out; the ampli- 
tude of the sine-curve will gradually diminish till it disappears ; 
in short, our phenomenon is simply expressed by what is known 
as a “damped sine-curve.” Exactly the same thing occurs in 
man, though neither in his case nor in that of the fish have we 
sufficient data for its complete illustration. 

We can demonstrate the fact, however, in the case of man by 
the help of certain very interesting measurements which have 
been recorded by Daffner*, of the height of German cadets, 
measured at half-yearly intervals. 


Growth in height of German military Cadets, in half-yearly 
periods. (Daffner.) 


Increment in cm. 
Height in cent. : 


Number — a4" Winter Summer 
observed Age October April October 4-year 4-year Year 
12 11-12 139-4 141-0 143°3 1-6 2-3 3-9 
80 12-13 143-0 144-5 147-4 1-5 2-9 4-4 
146 13-14 147-5 149-5 152-5 2-0 3:0 5-0 
162 14-15 152-2 155-0 158-5 2-5 35 6-0 
162 15-16 158°5 160-8 163-8 2-3 3:0 5:3 
150 16-17 163-5 165-4 167-7 1-9 2:3 4-2 
82 17-18 167-7 168-9 170-4 1-2 1-5 2:7 
22 18-19 169-8 170-6 171-5 0:8 0-9 ley) 
6 19-20 170-7 171-1 171-5 0-4 0-4 0-8 


In the accompanying diagram (Fig. 30) the half-yearly incre- 
ments are set forth, from the above table, and it will be seen that 
they form two even and entirely separate series. The curve 
joing up each series of points is an acceleration-curve; and the 
comparison of the two curves gives a clear view of the relative 
rates of growth during winter and summer, and the fluctuation 
which these velocities undergo during the years in question. The 
dotted line represents, approximately, the acceleration-curve in 
its continuous fluctuation of alternate seasonal decrease and 
increase. 


In the case of trees, the seasonal fluctuations of growth} admit 


* Das Wachstum des Menschen, p. 329, 1902. 
+ The diurnal periodicity is beautifully shewn in the case of the Hop by Joh. 
Schmidt (C. R. du Laboratoire de Carlsberg, x, pp. 235-248, Copenhague, 1913). 


A 


111] THE EFFECT OF TEMPERATURE 119 


of easy determination, and it is a point of considerable interest 
to compare the phenomenon in evergreen and in deciduous trees. 
I happen to have no measurements at hand with which to make 
this comparison in the case of our native trees, but from a paper 
by Mr Charles E. Hall* I have compiled certain mean values for 
growth in the climate of Uruguay. 


Fig. 30. Half-yearly increments of growth, in cadets of various ages. 
(From Daffner’s data.) 


Mean monthly wncrease in Girth of Evergreen and Deciduous Trees, 
at San Jorge, Uruguay. (After C. BE. Hall.) Values expressed 
as percentages of total annual increase. 

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 
Hyersreens 9:0 -8:8 8:6 8:9 7-7 5-4 <43° 6-0 9: TDs 10-8 10-2 
Deciduous 
treet eee. 20-0 14-6 9:0 92-3 0-8 0:3, 0-44 ee Sar, a -9 16-7 221-0 

The measurements taken were those of the girth of the tree, 
in mm., at three feet from the ground. The evergreens included 
species of Pinus, Eucalyptus and Acacia; the deciduous trees 
included Quercus, Populus, Robinia and Melia. I have merely 
taken mean values for these two groups, and expressed the 
monthly values as percentages of the mean annual increase. The 
result (as shewn by Fig. 31) is very much what we might have 
expected. Th» growth of the deciduous trees is completely 
arrested in winter-time, and the arrest is all but complete over 


* Trans. Botan. Soc. Edinburgh, xvui, 1891, p. 456. 


120 THE RATE OF GROWTH [CH. 


a considerable period of time; moreover, during the warm season, 
the monthly values are regularly graded (approximately in a 
sine-curve) with a clear maximum (in the southern hemisphere) 
about the month of December. In the evergreen trees, on the 
other hand, the amplitude of the periodic wave is very much 
less; there is a notable amount of growth all the year round, 
and, while there is a marked diminution in rate during the coldest 
months, there is a tendency towards equality over a considerable 


of j 
20 


Deciduous 22 —22- 22 brees 


Evergreens 


Percentage of total annual growth 


0) 


t bn ee 
JF OM. CAR MSE oad As er Ol Noes 


Fig. 31. Periodic annual fluctuation in rate of growth of trees (in the 
southern hemisphere). 


part of the warmer season. It is probable that some of the 
species examined, and especially the pines, were definitely retarded 
in growth, either by a temperature above their optimum, or by 
deficiency of moisture, during the hottest period of the year; 
with the result that the seasonal curve in our diagram has (as it 
were) its region of maximum cut off. 

In the case of trees, the seasonal periodicity of growth is so 
well marked that we are entitled to make use of the phenomenon 
in a converse way, and to draw deductions as to variations in 


1] THE EFFECT OF CLIMATE 121 


climate during past years from the record of varying rates of 
growth which the tree, by the thickness of its annual rings, has 
preserved for us. Mr A. E. Douglass, of the University of 
Arizona, has made a careful study of this question*, and I have 
received (through Professor H. H. Turner of Oxford) some measure- 
ments of the average width of the successive annual rings in “‘yellow 
pine,’ 500 years old, from Arizona, in which trees the annual 
rings are very clearly distinguished. From the year 1391 to 1518, 
the mean of two trees was used; from 1519 to 1912, the mean of 
five; and the means of these, and sometimes of larger numbers, 
were found to be very concordant. A correction was applied by 
drawing a long, nearly straight line through the curve for the 
whole period, which line was assumed to represent the slowly 
diminishing mean width of ring accompanying the increase of 
size, or age, of the tree; and the actual growth as measured was 
equated with this diminishing mean. The figures used give, 
accordingly, the ratio of the actual growth in each year to the 
mean growth corresponding to the age or magnitude of the tree 
at that epoch. 

It was at once manifest that the rate of growth so determined 
shewed a tendency to fluctuate in a long period of between 100 and 
200 years. I then smoothed in groups of 100 (according to Gauss’s 
method) the yearly values, so that each number thus found 
represented the mean annual increase during a century: that is 
to say, the value ascribed to the year 1500 represented the average 
annual growth during the whole period between 1450 and 1550, 
and soon. These values give us a curve of beautiful and surprising 
smoothness, from which we seem compelled to draw the direct 
conclusion that the climate of Arizona, during the last 500 years, 
has fluctuated -with a regular periodicity of almost precisely 150 
years. Here again we should be left in doubt (so far as these 


* T had not received, when this was written, Mr Douglass’s paper, On a method 
of estimating Rainfall by the Growth of Trees, Bull. Amer. Geograph. Soc. XLv1, 
pp. 321-335, 1914. Mr Douglass does not fail to notice the long period here 
described; but he lays more stress on the occurrence of shorter cycles (of 11, 21 
and 33 years), well known to meteorologists. Mr Douglass is inclined (and I think 
rightly) to correlate the variations in growth directly with fluctuations in rainfall. 
that is to say with alternate periods of moisture and aridity; but he points out 
that the temperature curves (and also the sunspot curves) are markedly similar. 


‘OI6GT-OI8I “2'¥ 0F OGFI-OGET “A°V WoIT 
‘(sporsad avad-jO] UI poeygoows) seer, vuoOZIIy FO YAMOAS Fo oye UT UOTyeNJony powed-SuoyT “ZE “Sip 


=a 
+— 
——, 


O9sL Oost OOZL , 0091 OOSI Orvl 


CH. III] THE EFFECT OF CLIMATE 123 


observations go) whether the essential factor be a fluctuation of 
temperature or an alternation of moisture and aridity; but the 
character of the Arizona climate, and the known facts of recent 
years, encourage the belief that the latter is the more direct and 
more important factor. 

It has been often remarked that our common European trees, 
such for instance as the elm or the cherry, tend to have larger 
leaves the further north we go; but in this case the phenomenon 
is to be ascribed rather to the longer hours of daylight than to 
any difference of temperature*. The point is a physiological one, 
and consequently of little importance to us here+; the main point 
for the morphologist is the very simple one that physical or 
climatic conditions have greatly influenced the rate of growth. 
The case is analogous to the direct influence of temperature in 
modifying the colouration of organisms, such as certain butterflies. 
Now if temperature affects the rate of growth in strict uniformity, 
alike in all directions and in all parts or organs, its direct effect 
must be limited to the production of local races or varieties differing 
from one another in actual magnitude, as the Siberian goldfinch 
or bullfinch, for instance, differ from our own. But if there be 
even ever so little of a discriminating action in the enhancement 
of growth by temperature, such that it accelerates the growth of 
one tissue or one organ more than another, then it is evident that 
it must at once lead to an actual difference of racial, or even 
* specific’ form. 

It is not to be doubted that the various factors of climate 
have some such discriminating influence. The leaves of our 
northern trees may themselves be an instance of it; and we have, 


* Tt may well be that the effect is not due to light after all; but to increased 
absorption of heat by the soil, as a result of the long hours of exposure to the sun. 

+ On growth in relation to light, see Davenport, Exp. Morphology, 11, ch. xvii. 
In some cases (as in the roots of Peas), exposure to light seems to have no effect 
on growth; in other cases, as in diatoms (according to Whipple’s experiments, 
quoted by Davenport, 0, p. 423), the effect of light on growth or multiplication 
is well-marked, measurable, and apparently capable of expression by a logarithmic 
formula. The discrepancy would seem to arise from the fact that, while light- 
energy always tends to be absorbed by the chlorophyll of the plant, converted into 
chemical energy, and stored in the shape of starch or other reserve materials, the 
actual rate of growth depends on the rate at which these reserves are drawn on: 
and this is another matter, in which light-energy is no longer directly concerned. 


124 THE RATE OF GROWTH [cH. 


probably, a still better instance of it in the case of Alpine plants *, 
whose general habit is dwarfed, though their floral organs suffer 
little or no reduction. The subject, however, has been little 
investigated, and great as its theoretic importance would be to 
us, we must meanwhile leave it alone. 


Osmotic factors in growth. 


The curves of growth which we have now been studying ~ 
represent phenomena which have at least a two-fold interest, 
morphological and physiological. To the morphologist, who 
recognises that form is a “function” of growth, the important 
facts are mainly these: (1) that the rate of growth is an orderly 
phenomenon, with general features common to very various 
organisms, while each particular organism has its own character- 
istic phenomena, or “specific constants”; (2) that rate of growth 
varies with temperature, that is to say with season and with 
climate, and with various other physical factors, external and 
internal; (3) that it varies in different parts of the body, and 
according to various directions or axes: such variations being 
definitely correlated with one another, and thus giving rise to 
the characteristic proportions, or form, of the organism, and to 
the changes in form which it undergoes in the course of its 
development. But to the physiologist, the phenomenon suggests 
many other important considerations, and throws much light on 
the very nature of growth itself, as a manifestation of chemical 
and physical energies. 

To be content to shew that a certain rate of growth occurs in 
a certain organism under certain conditions, or to speak of the 
phenomenon as a “reaction” of the living organism to its environ- 
ment or to certain stimuli, would be but an example of that “lack 
of particularity +” in regard to the actual mechanism of physical 
cause and effect with which we are apt in biology to be too easily 
satisfied. But in the case of rate of growth we pass somewhat 


* Cf. for instance, Nageli’s classical account of the effect of change of habitat 
on Alpine and other plants: Sitzwngsber. Baier. Akad. Wiss. 1865, pp. 228-284. 

7 Cf. Blackman, F. F., Presidential Address in Botany, Brit. Ass. Dublin, 1908. 
The fact was first enunciated by Baudrimont and St Ange, Recherches sur le 
développement du foetus, Mém. Acad. Sci. xt, p. 469, 1851. 


111] OSMOTIC FACTORS IN GROWTH 125 


beyond these limitations; for the affinity with certain types of 
chemical reaction is plain, and has been recognised by a great 
number of physiologists. ; 

A large part of the phenomenon of growth, both in animals 
and still more conspicuously in plants, is associated with “turgor,” 
that is to say, is dependent on osmotic conditions; in other words, 
the velocity of growth depends in great measure (as we have already 
seen, p. 113) on the amount of water taken up into the living 
cells, as well as on the actual amount of chemical metabolism 
performed by them*. Of the chemical phenomena which result 
in the actual increase of protoplasm we shall speak presently, but 
the réle of water in growth deserves also a passing word, even in 
our morphological enquiry. 

It has been shewn by Loeb that in Cerianthus or Tubularia, 
for instance, the cells in order to grow must be turgescent; and 
this turgescence is only possible so long as the salt water in which 
the cells lie does not overstep a certain limit of concentration. The 
limit, in the case of Tubularia, is passed when the salt amounts 
to about 5:4 per cent. Sea-water contains some 3-0 to 3-5 p.c. 
of salts; but it is when the salinity falls much below this normal, 
to about 2-2 p.c., that Tubularia exhibits its maximal turgescence, 
and maximal growth. A further dilution is said to act as a poison 
to the animal. Loeb has also shewn jf that in certain eggs (e.g. 
those of the little fish Fundulus) an increasing concentration of 
the sea-water (leading to a diminishing “water-content” of the 
egg) retards the rate of segmentation and at length renders 
segmentation impossible; though nuclear division, by the way, 
goes on for some time longer. 

Among many other observations of the same kind, those of 
Bialaszewiezt, on the early growth of the frog, are notable. 
He shews that the growth of the embryo while still within the 


* Cf. Loeb, Untersuchungen zur physiol. Morphologie der Thiere, 1892; also 
Experiments on Cleavage, J. of Morph. vu, p. 253, 1892; Zusammenstellung der 
Ergebnisse einiger Arbeiten iiber die Dynamik des thierischen Wachsthum, Arch. 
f. Entw. Mech. xv, 1902-3, p. 669; Davenport, On the Réle of Water in Growth, 
_ Boston Soc. N, H. 1897; Ida H. Hyde, Am. J. of Physiol. x11, 1905, p. 241, ete. 

+ Pfliger’s Archiv, Lv, 1893. 

{ Beitrage zur Kenntniss der Wachstumsvorgange bei Amphibienembryonen, 
Bull. Acad. Sci. de Cracovie, 1908, p. 783; cf. Arch. f. Entw. Mech. xxvii, p. 160, 
1909; xxxiv, p. 489, 1912. 


126 THE RATE OF GROWTH [CH. 


vitelline membrane depends wholly on the absorption of water; 
that whether rate of growth be fast or slow (in accordance with 
temperature) the quantity of water absorbed is constant; and 
that successive changes of form correspond to definite quantities - 
of water absorbed. The solid residue, as Davenport has also 
shewn, may actually and notably diminish, while the embryo 
organism is increasing rapidly in bulk and weight. 

On the other hand, in later stages and especially in the higher 
animals, the percentage of water tends to diminish. This has 
been shewn by Davenport in the frog, by Potts in the chick, and 
particularly by Fehling in the case of man*. Fehling’s results 
are epitomised as follows: ; 

Age in weeks Bee: BG 17 22 24 26 30 35 = 339 

Percentage of water 97:5 91-8 92:0 89:9 86:4 83:7 82:9 74-2 


And the following illustrate Davenport's results for the frog: 


Age in weeks Snel 2 5 7 9 14 41 84 
Percentage of water 56:3 58-5 76-7 89-3 93-1 95-0 90-2 87-5 


To such phenomena of osmotic balance as the above, or in other 
words to the dependence of growth on the uptake of water, Héber + 
and also Loeb are inclined to refer the modifications of form 
which certain phyllopod crustacea undergo, when the highly 
saline waters which they inhabit are further concentrated, or are 
abnormally diluted. Their growth, according to Schmankewitsch, 
is retarded by increase of concentration, so that the individuals 
from the more saline waters appear stunted and dwarfish; and 
they become altered or transformed in other ways, which for the 
most part suggest “degeneration,” or a failure to attain full and 
perfect ‘development. Important physiological changes also 
ensue. The rate of multiplication is increased, and partheno- 
genetic reproduction is encouraged. Male individuals become 
plentiful in the less saline waters, and here the females bring forth 

* Febling, H., Arch. fiir Gynaekologie, x1, 1877; cf. Morgan, Hxperimental 
Zoclogy, p. 240, 1907. 

+ Hober, R., Bedeutung der Theorie der Loésungen fiir Physiologie und 
Medizin, Biol. Centralbl. xx, 1899; ef. pp. 272-274. 

t Schmankewitsch has made other interesting observations on change of size 


and form, after some generations, in relation to change of density; e.g. in the 
flagellate infusorian Anisonema acinus, Biitschli (Z. f. w. Z. xxix, p. 429, 1877). 


TI | OSMOTIC FACTORS IN GROWTH 127 


their young alive; males disappear altogether in the more con- 
centrated brines, and then the females lay eggs, which, however, 
only begin to develop when the salinity is somewhat reduced. 
The best-known case is the little “brine-shrimp,” Artemia 
salina, found, in one form or another, all the world over, and first 
discovered more than a century and a half ago in the salt-pans at 
Lymington. Among many allied forms, one, A. milhauseniv, 
inhabits the natron-lakes of Egypt and Arabia, where, under the 
name of “loul,” or “Fezzan-worm,” it is eaten by the Arabs*. 
This fact is interesting, because it indicates (and investigation 
has apparently confirmed) that the tissues of the creature are not 
impregnated with salt, as is the medium in which it lives. The 
fluids of the body, the mliew interne (as Claude Bernard called 
them +), are no more salt than are those of any ordinary crus- 
tacean or other animal, but contain only some 0:8 per cent. of 
NaClt, while the milew externe may contain 10, 20, or more per 
cent. of this and other salts; which is as much as to say that 
the skin, or body-wall, of the creature acts as a “semi-permeable 
membrane,” through which the dissolved salts are not permitted 
to diffuse, though water passes through freely: until a statical 
equilibrium (doubtless of a complex kind) is at length attained. 
Among the structural changes which result from increased 
concentration of the brine (partly during the life-time of the 
individual, but more markedly during the short season which 
suffices for the development of three or four, or perhaps more, 
successive generations), it 1s found that the tail comes to bear 
fewer and fewer bristles, and the tail-fins themselves tend at last 
to disappear; these changes corresponding to what have been 


>’ 


* These ‘‘Fezzan-worms,”’ when first described, were supposed to be “‘insects’ 
eges”; cf. Humboldt, Personal Narrative, vi, i, 8. note; Kirby and Spence. 
Letter x. 

+ Cf. Introd. a Vétude de la médecine expérimentale, 1885, p. 110. 

t Cf. Abonyi, Z. f. w. Z. oxtv, p. 134, 1915. But Frédéricq has shewn that 
the amount of NaCl in the blood of Crustacea (Carcinus moenas) varies, and 
all but corres onds, with the density of the water in which the creature 
has been kept (Arch. de Zool. Exp. et Gén. (2), U1, p. xxxv, 1885); and 
other results of Frédéricq’s, and various data given or quoted by Bottazzi 
(Osmotischer Druck und elektrische Leitungsfihigkeit der Fliissigkeiten der 
Organismen, in Asher-Spiro’s Ergebn. d. Physiologie, vit, pp. 160-402, 1908) suggest 
that the case of the brine-shrimps must be looked upon as an extreme or exceptional 
one. 


128 THE RATE OF GROWTH [oH. 


described as the specific characters of A. milhausenni, and of a 
still more extreme form, A. képpeniana; while on the other 
hand, progressive dilution of the water tends to precisely opposite 
conditions, resulting in forms which have also been described as 
separate species, and even referred to a separate genus, Callaonella, 
closely akin to Branchipus (Fig. 33). Pari passu with these changes, 
there is a marked change in the relative lengths of the fore and 
hind portions of the body, that is to say, of the “cephalothorax” 
and abdomen: the latter growing relatively longer, the salter the 
water. In other words, not only is the rate of growth of the whole 


YW Wy allay 
ae, 


— ees - Se fe —— SS 
ca = - ale pepe YS 
°: >. = = fa) ~ SS 
cS) ~ S Sango ates 
Ss > Se o mm & 
S 3 =. 8 = Ss. iS} 
= = 3 S = >= = 
Se cm a S a 
3 gS ~ % = 
5S =: a" Ey = 
= Sa =) 

— i a el Nae y 

Artemia s. str. Callaonella 


Fig. 33. Brine-shrimps (Artemia), from more or less saline water. Upper figures 
shew tail-segment and tail-fins; lower figures, relative length of cephalothorax 
and abdomen. (After Abonyi.) 


animal lessened by the saline concentration, but the specific rates 
of growth in the parts of its body are relatively changed. This 
latter phenomenon lends itself to numerical statement, and Abonyi 
has lately shewn that we may construct a very regular curve, by 
plotting the proportionate length of the creature’s abdomen 
against the salinity, or density, of the water; and the several 
species of Artemia, with all their other correlated specific characters, 
are then found to occupy successive, more or less well-defined, and 
more or less extended, regions of the curve (Fig. 33). In short, the 
density of the water is so clearly a “function” of the specific 


TI] OSMOTIC FACTORS IN GROWTH 129 


character, that we may briefly define the species Artemia (Callao- 
nella) Jelskw, for instance, as the Artemia of density 1000-1010 
(NaCl), or the typical A. salina, or principalis, as the Artemia 
of density 1018-1025, and so forth. It is a most interesting 
fact that these Artemiae, under the protection of their semi- 
permeable skins, are capable of living in waters not only of 
great density, but of very varied chemical composition. The 
natron-lakes, for instance, contain large quantities of magnesium 


K6ppeniana 


160 


140'— 


120 


Percentage ratio 


100 


80 al. Aes Ne IN cy eT | ie 
1000 1020 1040 1060 1080 2000 
Density of water 


Fig. 34. Percentage ratio of length of abdomen to cephalothorax in brine-shrimps, 
at various salinities. (After Abonyi.) 


sulphate; and the Artemiae continue to live equally well in 
artificial solutions where this salt, or where calcium chloride, has 
largely taken the place of sodium chloride in the more common 
habitat. Furthermore, such waters as those of the natron-lakes 
are subject to very great changes of chemical composition as 
concentration proceeds, owing to the different solubilities of the 
constituent salts. It appears that the forms which the Artemiae 
assume, and the changes which they undergo, are identical or 


DaGs 9 


130 THE RATE OF GROWTH [cH. 


indistinguishable, whichever of the above salts happen to exist, 
or to predominate, in their saline habitat. At the same time we 
still lack (so far as I know) the simple, but crucial experiments 
which shall tell us whether, in solutions of different chemical 
composition, it is at equal densities, or at “isotonic” concentrations 
(that is to say, under conditions where the osmotic pressure, 
and consequently the rate of diffusion, is identical), that the 
same structural changes are produced, or corresponding phases 
of equilibrium attained. 

While Hoéber and others* have referred all these phenomena to 
osmosis, Abonyi is inclined to believe that the viscosity, or 
mechanical resistance, of the fluid also reacts upon the organism ; 
and other possible modes of operation havé been suggested. 
But we may take it for certain that the phenomenon as a whole 
is not a simple one; and that it includes besides the passive 
phenomena of intermolecular diffusion, some other form of activity 
which plays the part of a regulatory mechanismy. 


Growth and catalytic action. 


In ordinary chemical reactions we have to deal (1) with a 
specific velocity proper to the particular reaction, (2) with varia- 
tions due to temperature and other physical conditions, (3) according 
to van’t Hofi’s ‘‘ Law of Mass,’’ with variations due to the actual 
quantities present of the reacting substances, and (4) in certain 
cases, with variations due to the presence of “catalysing agents.” 
In the simpler reactions, the law of mass involves a steady, gradual 
slowing-down of the process, according to a logarithmic ratio, as 
the reaction proceeds and as the initial amount of substance 
diminishes; a phenomenon, however, which need not necessarily 


* Cf. Schmankewitsch, Z. f. w. Zool. xxv, 1875, xx1x, 1877, etc.; transl. in 
appendix to Packard’s Monogr. of N. American Phyllopoda, 1883, pp. 466-514; 
Daday de Deés, Ann. Sci. Nat. (Zool.), (9), x1, 1910; Samter und Heymons, Abdh. 
d. K. pr. Akad. Wiss. 1902; Bateson, Mat. for the Study of Variation, 1894, pp. 
96-101; Anikin, Mitth. Kais. Univ. Tomsk, x1v: Zool. Centralbl. v1, pp. 756-760, 
1908; Abonyi, Z. f. w. Z. cxtv, pp. 96-168, 1915 (with copious bibliography), ete. 

+ According to the empirical canon of physiology, that (as Frédéricq expresses 
it) “L’étre vivant est agencé de telle maniere que chaque influence perturbatrice 
provoque d’elle-méme la mise en activité de Vappareil compensateur qui doit 
neutraliser et réparer le dommage.” 


111] GROWTH AND CATALYTIC ACTION 131 


occur in the organism, part of whose energies are devoted to the 
continual bringing-up of fresh supplies. 

Catalytic action occurs when some substance, often in very 
minute quantity, is present, and by its presence produces or 
accelerates an action, by opening ‘‘a way round,” without 
the catalytic agent itself being diminished or used  up%*. 
Here the velocity curve, though quickened, is not necessarily 
altered in form, for gradually the law of mass exerts its 
effect and the rate of the reaction gradually diminishes. But 
in certain cases we have the very remarkable phenomenon that 
a body acting as a catalyser is necessarily formed as a product, 
or bye-product, of the main reaction, and in such a case as this 
the reaction-velocity will tend to be steadily accelerated. Instead 
of dwinding away, the reaction will continue with an ever- 
increasing velocity: always subject to the reservation that limiting 
conditions will in time make themselves felt, such as a failure of 
some necessary ingredient, or a development of some substance 
which shall antagonise or finally destroy the original reaction. 
Such an action as this we have learned, from Ostwald, to describe 
as “autocatalysis.” Now we know that certain products of 
protoplasmic metabolism, such as the enzymes, are very powerful 
catalysers, and we are entitled to speak of an autocatalytic action 
on the part of protoplasm itself. This catalytic activity of pro- 
toplasm is a very important phenomenon. As Blackman says, 
in the address already quoted, the botanists (or the zoologists) 
“eall it growth, attribute it to a specific power of protoplasm for 
assimilation, anid leave it alone as a fundamental phenomenon ; 
but they are much concerned as to the distribution of new growth 
in innumerable specifically distinct forms.” While the chemist, on 
the other hand, recognises it as a familiar phenomenon, and refers it 
to the same category as his other known examples of autocatalysis. 


* Such phenomena come precisely under the head of what Bacon called 
Instances of Magic: ‘ By which I mean those wherein the material or efficient 
cause is scanty and small as compared with the work or effect produced; so that 
even when they are common, they seem like miracles, some at first sight, others 
even after attentive consideration. These magical effects are brought about in 
three ways...[of which one is] by excitation or invitation in another body, as in 
the magnet which excites numberless needles without losing any of its virtue, or 
an yeast and such-like.’ Nov. Org., cap. li. 


9—2 


132 THE RATE OF GROWTH [CH. 


This very important, and perhaps even fundamental pheno- 
menon of growth would seem to have been first recognised by 
Professor Chodat of Geneva, as we are told by his pupil Monnier *. 
“On peut bien, ainsi que M. Chodat l’a proposé, considérer 
Paccroissement comme une réaction chimique complexe, dans 
laquelle le catalysateur est la cellule vivante, et les corps en 
présence sont l’eau, les sels, et acide carbonique.” 

Very soon afterwards a similar suggestion was made by Loebt, 
in Gonnection with the synthesis of nuclein or nuclear protoplasm ; 
for he remarked that, as in an autocatalysed chemical reaction, 
the velocity of the synthesis increases during the initial stage of 
cell-division in proportion to the amount of nuclear matter already 
synthesised. In other words, one of the products of the reaction, 
i.e. one of the constituents of the ‘nucleus, accelerates the pro- 
duction of nuclear from cytoplasmic material. 

The phenomenon of autocatalysis is by no means confined to 
hving or protoplasmic chemistry, but at the same time it is 
characteristically, and apparently constantly, associated therewith. 
And it would seem that to it we may ascribe a considerable part 
of the difference between the growth of the organism and the 
simpler growth of the crystal{: the fact, for instance, that the cell 
can grow in a very low concentration of its nutritive solution, 
while the crystal grows only in a supersaturated one; and the 
fundamental fact that the nutritive solution need only contain 
the more or less raw materials of the complex constituents of the 
cell, while the crystal grows only in a solution of its own actual 
substance §. 

As F. F. Blackman has pointed out, the multiplication of an 
organism, for instance the prodigiously rapid increase of a bacterium, 


* Monnier, A., Les matiéres minérales, et la loi d’accroissement des Végétaux, 
Publ. de VInst. de Bot. de VUniv. de Genéve (7), 1, 1905. Cf. Robertson, On the 
Normal Rate of Growth of an Individual, and its Biochemical Significance, Arch. 
f. Entw. Mech. xxv, pp. 581-614, xxvi, pp. 108-118, 1908; Wolfgang Ostwald, 
Die zeitlichen Eigenschaften der Entwickelungsvorgdnge, 1908; Hatai, 8., Interpreta- 
tion of Growth-curves from a Dynamical Standpoint, Anat. Record, v, p. 373, 
1911. 

{+ Biochem. Zeitschr. 1, 1906, p. 34. 

{ Even a crystal may be said, in a sense, to display “autocatalysis”: for the 
bigger its surface becomes, the more rapidly does the mass go on increasing. 

§ Cf. Loeb, The Stimulation of Growth, Science, May 14, 1915. 


111] GROWTH AND CATALYTIC ACTION 133 


which tends to double its numbers every few minutes, till (were 
it not for limiting factors) its numbers would be all but incalculable 
in a day*,is a simple but most striking illustration of the potenti- 
alities of protoplasmic catalysis; and (apart from the large share 
taken by mere “turgescence” or imbibition of water) the same 
is true of the growth, or cell-multiplication, of a multicellular 
organism in its first stage of rapid acceleration. 

It is not necessary for us to pursue this subject much further, 
for it is sufficiently clear that the normal “curve of growth” of 
an organism, in all its general features, very closely resembles the 
velocity-curve of chemical autocatalysis. We see in it the first 
and most typical phase of greater and greater acceleration; this 
is followed by a phase in which limiting conditions (whose details 
are practically unknown) lead to a falling off of the former 
acceleration; and in most cases we come at length to a third phase, 
in which retardation of growth is succeeded by actual diminution 
of mass. Here we may recognise the influence of processes, or 
of products, which have become actually deleterious; their 
deleterious influence is staved off for awhile, as the organism draws 
on its accumulated reserves, but they lead ere long to the stoppage 
of all activity, and to the physical phenomenon of death. But 
when we have once admitted that the limiting conditions of 
growth, which cause a phase of retardation to follow a phase 
of acceleration, are very imperfectly known, it is plain that, 
ipso facto, we must admit that a resemblance rather than an 
identity between this phenomenon and that of chemical auto- 
catalysis is all that we can safely assert meanwhile. Indeed, as 
Enriques has shewn, points of contrast between the two phenomena 
are not lacking; for instance, as the chemical reaction draws to 
a close, it is by the gradual attainment of chemical equilibrium: 
but when organic growth draws to a close, it is by reason of a very 
different kind of equilibrium, due in the main to the gradual 
differentiation of the organism into parts, among whose peculiar 


* B. coli-communis, according to Buchner, tends to double in 22 minutes; in 
24 hours, therefore, a single individual would be multiplied by something like 
10°8; Sitzungsher. Miinchen. Ges. Morphol. u. Physiol. ut, pp. 65-71, 1888. Cf. 
Marshall Ward, Biology of Bacillus ramosus, etc. Pr. R. S. uvim, 265-468, 1895. 
The comparatively large infusorian Stylonichia, according to Maupas, would 
multiply in a month by 10%. 


134 THE RATE OF GROWTH [CH. 


and specialised functions that of cell-multiplication tends to fall 
into abeyance*. 

It would seem to follow, as a natural consequence, from what 
has been said, that we could without much difficulty reduce our 
curves of growth to logarithmic formulae} akin to those which 
the physical chemist finds applicable to his autocatalytic reactions. 
This has been diligently attempted by various writers{; but the 
results, while not destructive of the hypothesis itself, are only 
partially successful. The difficulty arises mainly from the fact 
that, in the life-history of an organism, we have usually to deal 
(as indeed we have seen) with several recurrent periods of relative 
acceleration and retardation. It is easy to find a formula which 
shall satisfy the conditions during any one of these periodic 
phases, but it is very difficult to frame a comprehensive formula 
which shall apply to the entire period of growth, or to the whole 
duration of life. 

But if it be meanwhile impossible to formulate or to solve in 
precise mathematical terms the equation to the growth of an 
organism, we have yet gone a very long way towards the solution . 
of such problems when we have found a “qualitative expression,” 
as Poincaré puts it; that is to say, when we have gained a fair 
approximate knowledge of the general curve which represents the 
unknown function. 


As soon as we have touched on such matters as the chemical 
phenomenon of catalysis, we are on the threshold of a subject 
which, if we were able to: pursue it, would soon lead us far into 
the special domain of physiology; and there it would be necessary 
to follow it if we were dealing with growth as a phenomenon in 
itself, instead of merely as a help to our study and comprehension 
of form. For instance the whole question of diet, of overfeeding 
and underfeeding, would present itself for discussion§. But 
without attempting to open up this large subject, we may say a 

* Cf. Enriques, Wachsthum und seine analytische Darstellung, Biol. Centralbl. 
1909, p. 337. 


+ CE. (ant. al.) Mellor, Chemical Statics and Dynamics, 1904, p. 291. 
+ Cf. Robertson, /.c. 


§ See, for a brief resumé of this subject, Morgan’s Haperimental Zoology, 
chap. Xvi. 


111] GROWTH AND CATALYTIC ACTION 135 


further passing word upon the essential fact that certain chemical 
substances have the power of accelerating or of retarding, or in 
some way regulating, growth, and of so influencing directly the 
morphological features of the organism. 

Thus lecithin has been shewn by Hatai*, Danilewsky} and 
others to have a remarkable power of stimulating growth in 
various animals; and the so-called “auximones,” which Professor 
Bottomley prepares by the action of bacteria upon peat appear 
to be, after a somewhat similar fashion, potent accelerators of 
the growth of plants. But by much the most interesting cases, 
from our point of view, are those where a particular substance 
appears to exert a differential effect, stimulating the growth of 
one part or organ of the body more than another. 

It has been known for a number of years that a diseased 
condition of the pituitary body accompanies the phenomenon 
known as “acromegaly,” in which the bones are variously enlarged 
or elongated, and which is more or less exemplified in every 
skeleton of a “giant”; while on the other hand, disease or extirpa- 
tion of the thyroid causes an arrest of skeletal development, and, 
if it take place early, the subject remains a dwarf. These, then, 
are well-known illustrations of the regulation of function by some 
internal glandular secretion, some enzyme or “hormone” (as 
Bayliss and Starling call it), or “harmozone,” as Gley calls it in 
the particular case where the function regulated is that of growth, 
with its consequent influence on form. 

Among other illustrations (which are plentiful) we have, for 
instance the growth of the placental decidua, which Loeb has 
shewn to be due to a substance given off by the corpus luteum 
of the ovary, giving to the uterine tissues an abnormal capacity 
for growth, which in turn is called into action by the contact of 
the ovum, or even of any foreign body. And various sexual 
characters, such as the plumage, comb and spurs of the cock, 
are believed in like manner to arise in response to some particular 
internal secretion. When the source of such a secretion is removed 
by castration, well-known morphological changes take place in 
various animals; and when a converse change takes place, the 
female acquires, in greater or less degree, characters which are 

* Amer. J. of Physiol., x, 1904. 7 C.R. cxxi, cxxi, 1895-96. 


136 THE RATE OF GROWTH [CH. 


proper to the male, as in certain extreme cases, known from time 
immemorial, when late in life a hen assumes the plumage of the 
cock. 

There are some very remarkable experiments by Gudernatsch, 
~in which he has shewn that by feeding tadpoles (whether of frogs 
or toads) on thyroid gland substance, their legs may be made to 
grow out at any time, days or weeks before the normal date of 
their appearance*. No other organic food was found to produce 
the same effect; but since the thyroid gland is known to contain 
iodine +, Morse experimented with this latter substance, and found 
that if the tadpoles were fed with iodised amino-acids the legs 
developed precociously, just as when the thyroid gland itself was 
used. We may take it, then, as an established fact, whose full 
extent and bearings are still awaiting investigation, that there 
exist substances both within and without the organism which 
have a marvellous power of accelerating growth, and of doing so 
in such a way as to affect not only the size but the form or pro- 
portions of the organism. 


If we once admit, as we are now bound to do, the existence 
of such factors as these, which, by their physiological activity 
and apart from any direct action of the nervous system, tend 
towards the acceleration of growth and consequent modification 
of form, we are led into wide fields of speculation by an easy and 
a legitimate pathway. Professor Gley carries such speculations 
a long, long way: for he sayst that by these chemical influences 
“Toute une partie de la construction des étres parait s expliquer 
dune facon toute mécanique. La forteresse, si longtemps inacces- 
sible, du vitalisme est entamée. Car la notion morphogénique 
était, suivant le mot de Dastre§$, comme ‘le dernier réduit de la 
force vitale.’” 

The physiological speculations we need not discuss: but, to 
take a single example from morphology, we begin to understand 
the possibility, and to comprehend the probable meaning, of the 

* Cf. Loeb, Science, May 14, 1915. 

+ Cf. Baumann u. Roos, Vorkommen von Iod im Thierkérper, Zeitschr. fiir 
Physiol. Chem. Xx1, xxut, 1895, 6. 


t Le Néo-Vitalisme, Rev. Scientifique, Mars 1911, p. 22 (of reprint). 
§ La vie et la mort, p. 43, 1902. 


111] GROWTH AND CATALYTIC ACTION 137 


all but sudden appearance on the earth of such exaggerated and 
almost monstrous forms as those of the great secondary reptiles 
and the great tertiary mammals*. We begin to see that it is in 
order to account, not for the appearance, but for the disappearance 
of such forms as these that natural selection must be invoked. 
And we then, I think, draw near to the conclusion that what is 
true of these is universally true, and that the great function of 
natural selection is not to originate, but to remove: donec ad 
interitum genus id natura redegitt. 

The world of things living, like the world of things inanimate, 
grows of itself, and pursues its ceaseless course of creative evolution. 
It has room, wide but not unbounded, for variety of living form 
and structure, as these tend towards their seemingly endless, but 
yet strictly limited, possibilities of permutation and degree: it 
has room for the great and for the small, room for the weak and 
for the strong: Environment and circumstance do not always 
' make a prison, wherein perforce the organism must either live 
or die; for the ways of life may be changed, and many a refuge 
found, before the sentence of unfitness is pronounced and the 
penalty of extermination paid. But there comes a time when 
“variation, in form, dimensions, or other qualities of the organism, 
goes farther than is compatible with all the means at hand of 
health and welfare for the individual and the stock; when, under 
the active and creative stimulus of forces from within and from 
without, the active and creative energies of growth pass the 
bounds of physical and physiological equilibrium: and so reach 
the limits which, as again Lucretius tells us, natural law has set 
between what may and what may not be, 


“et quid quaeque queant per foedera naturai 
quid porro nequeant.” 
Then, at last, we are entitled to use the customary metaphor, 
and to see in natural selection an inexorable force, whose function 


* Cf. Dendy, Evolutionary Biology, 1912, p. 408; Brit. Ass. Report (Portsmouth), 
1911, p. 278. 

+ Lucret. v. 877. “Lucretius nowhere seems to recognise the possibility of 
improvement or change of species by ‘natural selection’; the animals remain as 
they were at the first, except that the weaker and more useless kinds have been 
crushed out. Hence he stands in marked contrast with modern evolutionists.” 
Kelsey’s note, ad loc. 


138 THE RATE OF GROWTH lon. 


is not to create but to destroy,—to weed, to prune, to cut down 
and to cast into the fire*. 


Regeneration, or growth and repair. 


The phenomenon of regeneration, or the restoration of lost or 
amputated parts, is a particular case of growth which deserves 
separate consideration.. As we are all aware, this property is 
manifested in a high degree among invertebrates and many cold- 
blooded vertebrates, diminishing as we ascend the scale, until at 
length, in the warm-blooded animals, it lessens down to no more 
than that vis medicatricx which heals a wound. Ever since the 
days of Aristotle, and especially since the experiments of Trembley, 
Réaumur and Spallanzani in the middle of the eighteenth century, 
the physiologist and the psychologist have alike recognised that 
‘the phenomenon is both perplexing and important. The general 
phenomenon is amply discussed elsewhere, and we need, only 
deal with it in its immediate relation to growthf. 

Regeneration, like growth in other cases, proceeds with a 
velocity which varies according to a definite law; the rate varies 
with the time, and we may study it as velocity and as acceleration. 

Let us take, as an instance, Miss M. L. Durbin’s measurements 
of the rate of regeneration of tadpoles’ tails: the rate being here 
measured in terms, not of mass, but of length, or longitudinal 
increment {. 

From a number of tadpoles, whose average length was 34-2 mm., 
their tails being on an average 21-2 mm. long, about half the tail 


* Even after we have so narrowed the scope and sphere of natural selection, 
it is still hard to understand; for the causes of extinction are often wellnigh as hard ~ 
to comprehend as are those of the origin of species. If we assert (as has been 
lightly done) that Smilodon perished owing to its gigantic tusks, that Teleosaurus 
was handicapped by its exaggerated snout, or Stegosaurus weighed down by its 
intolerable load of armour, we may be reminded of other kindred forms to show 
that similar conditions did not necessarily lead to extermination, or that rapid 
extinction ensued apart from any such visible or apparent disadvantages. Cf. 
Lucas, F. A., On Momentum in Variation, Amer. Nat. xli, p. 46, 1907. 

+ See Professor T. H. Morgan’s Regeneration (316 pp.), 1901 for a full account 
and copious bibliography. The early experiments on regeneration, by Vallisneri, 
Réaumur, Bonnet, Trembley, Baster, and others, are epitomised by Haller, Elem. 
Physioloyiae, vat, p. 156 seq. 

t Journ. Experim. Zool. vit, p. 397, 1909. 


mt] REGENERATION, OR GROWTH AND REPAIR 139 


(11-5 mm.) was cut off, and the amounts regenerated in successive 
periods are shewn as follows: 


Days after operation... ee 7 10 14 18 24 #30 
(1) Amount regeneratedinmm. 14 34 43 52 55 62 6:5 
(2) Incrementduringeachperiod 14 2:0 09 09 O38 O7 O03 
(3)(?) Rate per day during 

each period a6e .-- 0:46 0:50 0:30 ° 0-25 0-07 0-12 0-05 


The first line of numbers in this table, if plotted as a curve 
against the number of days, will give us a very satisfactory view 
of the “curve of growth” within the period of the observations: 
that is to say, of the successive relations of length to time, or the 
velocity of the process. But the third line is not so satisfactory, 
and must not be plotted directly as an acceleration curve. For 
it is evident that the “rates” here determined do not correspond 
to velocities at the dates to which they are referred, but are the 
mean velocities over a preceding period; and moreover the periods 
over which these means are taken are liere of very unequal length. 
But we may draw a good deal more information from this experi- 
ment, if we begin by drawing a smooth curve, as nearly as possible 
through the points corresponding to the amounts regenerated 
(according to the first line of the table); and if we then interpolate 
from this smooth curve the actual lengths attained, day by 
day, and derive from these, by subtraction, the successive daily 
increments, which are the measure of the daily mean velocities 
(Table, p. 141). (The more accurate and strictly correct method 
would be to draw successive tangents to the curve.) 

In our curve of growth. (Fig. 35) we cannot safely interpolate 
values for the first three days, that is to say for the dates between 
amputation and the first actual measurement of the regenerated 
part. What goes on in these three days is very important; but 
we know nothing about it, save that our curve descended to zero 
somewhere or other within that period. As we have already 
learned, we can more or less safely interpolate between known 
points, or actual observations; but here we have no known 
starting-point. In short, for all that the observations tell us, 
and for all that the appearance of the curve can suggest, the 
curve of growth may have descended evenly to the base-line, 
which it would then have reached about the end of the second 


140 THE RATE OF GROWTH [CH. 


day; or it may have had within the first three days a change of 
direction, or “point of inflection,” and may then have sprung 


71 - + : - a 

cm. | 
growth 
6F 


a 


#4 ! ; ? ! ; : , ‘ 
0) ON 49 6 Sa 12 14 TEN 18) LO Neon e4 Poon ease 
; days 


Fig. 35. Curve of regenerative growth in tadpoles’ tails. (From 
M. L. Durbin’s data.) 


at once from the base-line at zero. That is to say, there may 
have been an intervening “latent period,’ during which no growth 


TT aah, -- T T =e T T n T 1 


cm! per day 


O 


9 4 6 8 10 12 14 16 18 20 29 24 26 2830 
days 


Fig. 36. Mean daily increments, corresponding to Fig. 35. 


m1] REGENERATION, OR GROWTH AND REPAIR 141 


occurred, between the time of injury and the first measurement 
of regenerative growth; or, for all we yet know, regeneration 
may have begun at once, but with a velocity much less than that 
- which it afterwards attained. This apparently trifling difference 
would correspond to a very great difference in the nature of the 
phenomenon, and would lead to a very striking difference in the 
curve which we have next to draw. . 

The curve already drawn (Fig. 35) illustrates, as we have seen, 
the relation of length to time, i.e. L/T = V. The second (Fig. 36) 
represents the rate of change of velocity; it sets V against T; 


The foregoing table, extended by graphic interpolation. 


Total Daily 
Days increment increment Logs of do. 
1) eek 
9 — 
3 1-40 
60 1-78 
4 2-00 ¢ 
52 1-72 
5 2-52 45 1-65 
oO ord) 
6 2:97 a6 
43 1-63 
7 3-40 ik 
nee 32 1-51 
8 3°72 
30 1-48 
9 4-02 2 
. 28 1-45 
10 4-30 aS é 
22 1-34 
11 4-52 ¢ ‘ 
é 21 1-32 
12 4:73 
19 1-28 
13 4-92 
18 1-26 
14 5:10 9 
ei iy, 1-23 
15 5:27 
13 Est 
16 5:40 
14 1-15 
17 5:54 
13 Ett 
18 5°67 
11 1-04 
19 5:78 
10 1-00 
20 5°88 
10 1-00 
21 5-98 
09 95 
22 6-07 
O7 85 
23 6-14 
07 “84 
24 6-21 08 90 
25 6-29 06 78 
26 6:35 2 
“06 78 
27 6-41 5 
05 ‘70 
28 6-46 
29 6-50 ie ae 
‘03 48 


142 THE RATE OF GROWTH [CH. 


and V/T or L/T?, represents (as we have learned) the acceleration of 
growth, this being simply the “differential coefficient,” the first 
derivative of the former curve. 

Now, plotting this acceleration curve from the date of the - 
first measurement made three days after the amputation of the 
tail (Fig. 36), we see that it has no point of inflection, but falls 
steadily, only more and more slowly, till at last it comes down 
nearly to the base-line. The velocities of growth are continually 
diminishing. As regards the missing portion at the beginning of 
the curve, we cannot be sure whether it bent round and came down 


1-0/- Ne 7 


i L 1 i | rl 1 i ee ae eee 
2 4 6 8 10 12 14 16 18 20 22 24 96 98 30 
days 


Fig. 37. Logarithms of values shewn in Fig. 36. 


to zero, or whether, as in our ordinary acceleration curves of growth 
from birth onwards, it started from a maximum. The former is, 
in this case, obviously the more probable, but we cannot be sure. 

As regards that large portion of the curve which we are 
acquainted with, we see that it resembles the curve known as 
a rectangular hyperbola, which is the form assumed when two 
variables (in this case V and 7) vary inversely as one another. 
If we take the logarithms of the velocities (as given in the table) 
and plot them against time (Fig. 37), we see that they fall, approxi- 
mately, into a straight line; and if this curve be plotted on the 


m1] REGENERATION, OR GROWTH AND REPAIR 143 


proper scale we shall find that the angle which it makes with the 
base is about 25°, of which the tangent is -46, or in round numbers $. 

Had the angle been 45° (tan 45° = 1), the curve would have 
been actually a rectangular hyperbola, with V7 = constant. As 
it is, we may assume, provisionally, that it belongs to the same 
family of curves,so that V™ 7”, or V™/"T, or VT”!™, are all severally 
constant. In other words, the velocity varies inversely as some 
power of the time, or vice versa. And in this particular case, the 
equation V7? = constant, holds very nearly true; that is to say 
the velocity varies, or tends to vary, inversely as the square of 


7 lee T mae T mrs T T T =T T =; | 


ote ae en B= 10--, 12. si aun omnes 
days 
Fig. 38. Rate of regenerative growth in larger tadpoles. 


the time. If some general law akin to this could be established 
as a general law, or even as a common rule, it would be of great 
importance. 

But though neither in this case nor in any other can the minute 
increments of growth during the first few hours, or the first couple 
of days, after injury, be directly measured, yet the most important 
point is quite capable of solution. What the foregoing curve 
leaves us in ignorance of, is simply whether growth starts at zero, 
with zero velocity, and works up quickly to a maximum velocity 
from which it afterwards gradually falls away; or whether after 
a latent period, it begins, so to speak, in full force. The answer 


144 THE RATE OF GROWTH [CH. 


to this question-depends on whether, in the days following the 
first actual measurement, we can or cannot detect a daily ancrement 
in velocity, before that velocity begins its normal course of diminu- : 
tion. Now this preliminary ascent to a maximum, or point of 
inflection of the curve, though not shewn in the above-quoted 
experiment, has been often observed: as for instance, in another 
similar experiment by the author of the former, the tadpoles being 
in this case of larger size (average 49-1 mm.)*. 

Days 3 5 7 10 12 14 17 24. - 280 Pese 
Increment 0:86 2-15 3:66 5:20 5:95 6:38 7:10 7-60 8-20 8-40 


Or, by graphic interpolation, 


Total Daily Total Daily 

Days increment do. Days increment do. 
i 23 23 10 5:29 39 
2 53 30 11 5°62 33 
3. “86 33 ate 5:90 28 
4 1-30 44 13 6-13 23 
5 2-00 “70 14 6:38 25 
6 . 2°78 "78 15 6-61 -23 
7 3°58 80 16 6-81 20 
8 4-30 “72 17 7-00 -19 ete. 
2 4-90 60 


The acceleration curve is drawn in Fig. 39. 

Here we have just what we lacked in the former case, namely 
a visible point of inflection in the curve about the seventh day 
(Figs. 38, 39), whose existence is confirmed by successive observa- 
tions on the 3rd, 5th, 7th and 10th days, and which justifies to 
some extent our extrapolation for the otherwise unknown period 
up to and ending with the third day; but even here there is a 
short space near the very beginning during which we are not 
quite sure of the precise slope of the curve. 


We have now learned that, according to these experiments, 
with which many others are in substantial agreement, the rate of 
growth in the regenerative process is as follows. After a very 
short latent period, not yet actually proved but whose existence 
is highly probable, growth commences with a velocity which very 


* Op. cit. p. 406, Exp. Iv. 


mI] REGENERATION, OR GROWTH AND REPAIR 145 


rapidly increases to a maximum. The curve quickly,—almost 
suddenly,—changes its direction, as the velocity begins to fall; 
and the rate of fall, that is, the negative acceleration, proceeds 
at a slower and slower rate, which rate varies inversely as some 
power of the time, and is found in both of the above-quoted 
experiments to be very approximately as 1/T?. But it is obvious 
that the value which we have found for the latter portion of the 
curve (however closely it be conformed to) is only an empirical 
value; it has only a temporary usefulness, and must in time give 


ie) T T Ta T cage igreily ST 
cm.|per day 


{3} 


r : ’ 1 ' [eee ay 1 are ce 
10) D 4 6 8 10 12 14 16 18 
; 2 days 


Fig. 39. Daily increment, or amount regenerated, corresponding to Fig, 38. 
4 


place to a formula which shall represent the entire phenomenon, 
from start to finish. 

While the curve of regenerative growth is apparently different 
from the curve of ordinary growth as usually drawn (and while 
this apparent difference has been commented on and treated as 
valid by certain writers) we are now in a position to see that it 
only looks different because we are able to study it, if not from 
the beginning, at least very nearly so: while an ordinary curve 
of growth, as it is usually presented to us, is one which dates, not 


T. G. 10 


146 THE RATE OF GROWTH [CH. 


from the beginning of growth, but from the comparatively late, 
and unimportant, and even fallacious epoch of birth. A complete 
curve of growth, starting from zero, has the same essential charac- 
teristics as the regeneration curve. 

Indeed the more we consider the phenomenon of regeneration, 
the more plainly does it shew itself to us as but a particular case 
of the general phenomenon of growth*, following the same lines, 
obeying the same laws, and merely started into activity by the 
special stimulus, direct or indirect, caused by the infliction of a 
wound. Neither more nor less than in other problems of physiology 
are we called upon, in the case of regeneration, to indulge in 
metaphysical speculation, or to dwell upon the beneficent purpose 
which seemingly underlies this process of healing and restoration. 


It is a very general rule, though apparently not a universal 
one, that regeneration tends to fall somewhat short of a complete 
restoration of the lost part; a certain percentage only of the lost 
tissues 1s restored. This fact was well known to some of those 
old investigators, who, like the Abbé Trembley and hke Voltaire, 
found a fascination in the study of artificial injury and the regenera- 
tion which followed it. Sir John Graham Dalyell, for instance, 
says, in the course of an admirable paragraph on regeneration T : 
“The reproductive faculty...is not confined to one portion, but 
may extend over many; and it may ensue even in relation to the 
regenerated portion more than once. Nevertheless, the faculty 
gradually weakens, so that in general every successive regeneration 
is smaller and more imperfect than the organisation preceding it; 
and at length it is exhausted.” 

In certain minute animals, such as the Infusoria, in which the 
capacity for “regeneration” is so great that the entire animal 
may be restored from the merest fragment, it becomes of great 
interest to discover whether there be some definite size at which 
the fragment ceases to display this power. This question has 

* The experiments of Loeb on the growth of Tubulazia in various saline 
solutions, referred to on p. 125, might as well or better have been referred to under 
the heading of regeneration, as they were performed on cut pieces of the zoophyte. 
(Cf. Morgan, op. cil. p. 35.) 


+ Powers of the Creator, 1, p. 7, 1851. See also Rare and Remarkable Animals, 
, pp. 17-19, 90, 1847. 


m1] REGENERATION, OR GROWTH AND REPAIR 147 


been studied by Lillie*, who found that in Stentor, while still 
smaller fragments were capable of surviving for days, the smallest 
portions capable of regeneration were of a size equal to a sphere of 
about 80 u in diameter, that is-to say of a volume equal to about 
one twenty-seventh of the average entire animal. He arrives at 
the remarkable conclusion that for this, and for all other species 
of animals, there is a “minimal organisation mass,” that is to say 
a “minimal mass of definite size consisting of nucleus and cyto- 
plasm within which the organisation of the species can just find 
its latent expression.”’ And in like manner, Boverit has shewn 
that the fragment of a sea-urchin’s egg capable of growing up into 
a new embryo, and so discharging the complete functions of an 
entire and uninjured ovum, reaches its limit at about one-twentieth 
of the original egg,—other writers having found a limit at about 
one-fourth. These magnitudes, small as they are, represent 
objects easily visible under a low power of the microscope, and so 
stand in a very different category to the minimal magnitudes in 
which life itself can be manifested, and which we have discussed 
in chapter IT. 

A number of phenomena connected with the linear rate of 
regeneration are illustrated and epitomised in the accompanying 
diagram (Fig. 40), which I have constructed from certain data 
given by Ellis in a paper on the relation of the amount of tail 
regenerated to the amount removed, in Tadpoles. These data are 
summarised in the next Table. The tadpoles were all very much 
of a size, about 40 ynm.; the average length of tail was very near 
to 26 mm., or 65 per cent. of the whole body-length; and in four 
series of experiments about 10, 20, 40 and 60 per cent. of the tail 
were severally removed. The amount regenerated in successive 
intervals of three days is shewn in our table. By plotting the 
actual amounts regenerated against these three-day intervals of 
time, we may interpolate values for the time taken to regenerate 
definite percentage amounts, 5 per cent., 10 per cent., etc. of the 

* Lillie, F. R., The smallest Parts of Stentor capable of Regeneration, 
Journ. of Morphology, xu, p. 239. 1897. 

+ Boveri, Entwicklungsfahigkeit kernloser Seeigeleier, etce., Arch. f. Entw. Mech. 
tm, 1895. See also Morgan, Studies of the partial larvae of Sphaerechinus, ibid. 


1895; J. Loeb, On the Limits of Divisibility of Living Matter, Biol. Lectures, 1894, 
etc. 


10—2 


148 THE RATE OF GROWTH [CH. 


The Rate of Regenerative Growth in Tadpoles’ Tails. 
(After M. M. Ellis, J. Exp. Zool. vit, p. 421, 1909.) 


Body Tail Amount Pei cent. % amount regenerated in days 

length length removed of tail —--t+"—_ 
Series* mm. mm. mm. removed ~3'| 6°" 9) 12) laa 
O 39-575 25-895 3-2 12°36 13 31 44 44 44 44 44 
PD 40-21 26-13 5:28 20:20 10 29 40 44 44 44 44 


R 39-86 25-70 10-4 40-50 6 20 31 40 48 48 48 
S 40-34 26-11 14-8 56-7 0 16 33 39 45 48 48 


* Each series gives the mean of 20 experiments. 


80 


removed 


1 


40 


er Lo A 6 ge 10. 12 | Oda ea al Gas 


2 
days 


Fig. 40. Relation between the percentage amount of tail removed, the percentage 
restored, and the time required for its restoration. (From M. M. Ellis’s 
data.) 


amount removed; and my diagram is constructed from the four 
sets of values thus obtained, that is to say from the four sets of 
experiments which differed from one another in the amount of 
tail amputated. To these we have to add the general result of a 
fifth series of experiments, which shewed that when as much as 
75 per cent. of the tail was cut off, no regeneration took place at 
all, but the animal presently died. In our diagram, then, each 


m1] REGENERATION, OR GROWTH AND REPAIR 149 


curve indicates the time taken to regenerate n per cent. of the 
amount removed. All the curves converge towards infinity, when 
the amount removed (as shewn by the ordinate) approaches 75 
per cent.; and all of the curves start from zero, for nothing is 
regenerated where nothing had been removed. Hach curve ap- 
proximates in form to a cubic parabola. 

The amount regenerated varies also with the age of the tadpole 
and with other factors, such as temperature; in other words, for 
any given age, or size, of tadpole and also for various, specific 
temperatures, a similar diagram might be constructed. 


The power of reproducing, or regenerating, a lost limb is 
particularly well developed in arthropod animals, and is some- 
times accompanied by remarkable modification of the form of 
the regenerated limb. A case in point, which has attracted 
much attention, occurs in connection with the claws of certain 
Crustacea *. 

In many Crustacea we have an asymmetry of the great claws, 
one being larger than the other and also more or less different in 
form. For instance, in the common lobster, one claw, the larger 
of the two, is provided with a few great “crushing” teeth, while 
the smaller claw has more numerous teeth, small and serrated. 
Though Aristotle thought otherwise, it appears that the crushing- 
claw may be on the right or left side, indifferently; whether it 
be on one or the other is a problem of “‘chance.” It is otherwise 
in many other Crustacea, where the larger and more powerful 
claw is always left or right, as the case may be, according to the 
species: where, in other words, the “probability” of the large 
or the small claw being left or being right is tantamount to 
certaintyT. 

The one claw is the larger because it has grown the faster ; 


* Of. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, Arch. f. Entw. 
Mech. x1x, 181-247, 1905; xxv, 266-344, 1907. Emmel, ibid. xxm, 542, 1906; 
Regeneration of lost parts in Lobster, Rep. Comm. Inland Fisheries, Rhode Island, 
XXXV, xxxvi, 1905-6; Science (n.s.), xxvi, 83-87, 1907. Zeleny, Compensatory 
Regulation, J. Hxp. Zool. 1, 1-102, 347-369, 1905; etc. 

+ Lobsters are occasionally found with two symmetrical claws: which are then 
‘ usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. Calman, 
P.Z.S. 1906, pp. 633, 634, and reff. 


150 THE RATE OF GROWTH [CH. 


6 


it has a higher “coefficient of growth,” and accordingly, as age 
advances, the disproportion between the two claws becomes more 
and more evident. Moreover, we must assume that the character- 
istic form of the claw is a “function” of its magnitude; the 
knobbiness is a phenomenon coincident with growth, and we 
never, under any circumstances, find the smaller claw with big 
crushing teeth and the big claw with little serrated ones. There 
are many other somewhat similar cases where size and form are 
manifestly correlated, and we have already seen, to some extent, 
that thé phenomenon of growth is accompanied by certain ratios 
of velocity that lead inevitably to changes of form. Meanwhile, 
then, we must simply assume that the essential difference between 
the two claws is one of magnitude, with which a certain differentia- 
tion of form is inseparably associated. 

If we amputate a claw, or if, as often happens, the crab “casts 
it off,” it undergoes a process of regeneration,—it grows anew, 
and evidently does so with an accelerated velocity, which accelera- 
tion will cease when equilibrium of the parts is once more attained : 
the accelerated velocity being a case in point to illustrate that 
vis revulsionis of Haller, to which we have already referred. 

With the help of this principle, Przibram accounts for certain 
curious phenomena which accompany the process of regeneration. 
As his experiments and those of Morgan shew, uf the large or 
knobby claw (A) be removed, there are certain cases, e.g. the 
common lobster, where it is directly regenerated. In other cases, 
e.g. Alpheus*, the other claw (B) assumes the size and form of that 
which was amputated, while the latter regenerates itself in the 
form of the other and weaker one; A and B have apparently 
changed places. In a third case, as in the-crabs, the A-claw re- 
generates itself as a small or B-claw, but the B-claw remains for a 
time unaltered, though slowly and in the course of repeated moults 
it later on assumes the large and heavily toothed A-form. 

Much has been written on this phenomenon, but in essence it 
is very simple. It depends upon the respective rates of growth, 
upon a ratio between the rate of regeneration and the rate of 
growth of the uninjured limb: complicated a little, however, by 


* Wilson, E. B., Reversal of Symmetry in Alpheus heterochelis, Biol. Bull. tv, 
p. 197, 1903. 


11] REGENERATION, OR GROWTH AND REPAIR 151 


the possibility of the uninjured limb growing all the faster for 
a time after the animal has been relieved of the other. From the 
time of amputation, say of A, A begins to grow from zero, with 
a high “regenerative” velocity; while B, starting from a definite 
magnitude, continues to increase, with its normal or perhaps 
somewhat accelerated velocity. The ratio between the two 
velocities of growth will determine whether, by a given time, 
A has equalled, outstripped, or still fallen short of the magnitude 
of B. 

That this is the gist of the whole problem is confirmed (if 
confirmation be necessary) by certain experiments of Wilson’s. 
It is known that by section of the nerve to a crab’s claw, its 
growth is retarded, and as the general growth of the animal 
proceeds the claw comes to appear stunted or dwarfed. Now in 
such a case as that of Alpheus, we have seen that the rate of 
regenerative growth in an amputated large claw fails to let it 
reach or overtake the magnitude of the growing little claw: 
which latter, in short, now appears as the big one. But if at the 
same time as we amputate the big claw we also sever the nerve 
to the lesser one, we so far slow down the latter’s growth that 
the other is able to make up to it, and in this case the two claws 
continue to grow at approximately equal rates, or in other words 
continue of coequal size. 


The phenomenon of regeneration goes some way towards 
helping us to comprehend the phenomenon of “multiplication by 
fission,’ as it is exemplified at least in its simpler cases in many 
worms and worm-like animals. For physical reasons which we 
shall have to study in another chapter, there is a natural tendency 
for any tube, if it have the properties of a fluid or semi-fluid 
substance, to break up into segments after it comes to a certain 
length; and nothing can prevent its doing so, except the presence 
of some controlling force, such for instance as may be due to the 
pressure of some external support, or some superficial thickening 
or other intrinsic rigidity of its own substance. If we add to this 
natural tendency towards fission of a cylindrical or tubular worm, 
the ordinary phenomenon of regeneration, we have all that is 
essentially implied in “reproduction by fission.” And in so far 


152 THE RATE OF GROWTH [on 


as the process rests upon a physical principle, or natural tendency, 
we may account for its occurrence in a great variety of animals, 
zoologically dissimilar; and also for its presence here and absence 
there, in forms which, though materially different in a physical 
sense, are zoologically speaking very closely allied. 


CONCLUSION AND SUMMARY. 


But the phenomena of regeneration, like all the other 
phenomena of growth, soon carry us far afield, and we must draw 
this brief discussion to a close. 

For the main features which appear to be common to all 
curves of growth we may hope to have, some day, a physical 
explanation. In particular we should like to know the meaning 
of that point of inflection, or abrupt change from an increasing 
to a decreasing velocity of growth which all our curves, and 
especially our acceleration curves, demonstrate the existence of, 
provided only that they include the initial stages of the whole 
phenomenon: just as we should also like to have a full physical 
or physiological explanation of the gradually diminishing velocity 
of growth which follows, and which (though subject to temporary 
interruption or abeyance) is on the whole characteristic of growth in 
all cases whatsoever. In short, the characteristic form of the curve 
of growth in length (or any other linear dimension) is a phenomenon 
which we are at present unable to explain, but which presents 
us with a definite and attractive problem for future solution. 
It would seem evident that the abrupt change in velocity must be 
due, either to a change in that pressure outwards from within, 
by which the “forces of growth” make themselves manifest, or 
to a change in the resistances against which they act, that is to 
say the tension of the surface; and this latter force we do not by 
any means limit to “‘surface-tension” proper, but may extend to 
the development of a more or less resistant membrane or “skin,” 
or even to the resistance of fibres or other histological elements, 
binding the boundary layers to the parts within. I take it that 
the sudden arrest of velocity is much more likely to be due to a 
sudden increase of resistance than to a sudden diminution of 
internal energies: in other words, I suspect that it is coincident 
with some notable event of histological differentiation, such as 


rit] CONCLUSION AND SUMMARY 153 


the rapid formation of a comparatively firm skin; and that the 
dwindling of velocities, or the negative acceleration, which follows, 
is the resultant or composite effect of waning forces of growth on 
the one hand, and increasing superficial resistance on the other. 
This is as much as to say that growth, while its own energy tends 
to increase, leads also, after a while, to the establishment of 
resistances which check its own further increase, 

Our knowledge of the whole complex phenomenon of growth 
is so scanty that it may seem rash to advance even this tentative 
suggestion. But yet there are one or two known facts which 
seem to bear upon the question, and to indicate at least the manner 
in which a varying resistance to expansion may aftect the velocity 
of growth. For instance, it has been shewn by Frazee* that 
electrical stimulation of tadpoles, with small current density and 
low voltage, increases the rate of regenerative growth. As just 
such an electrification would tend to lower the surface-tension, 
and accordingly decrease the external resistance, the experiment 
would seem to support, in some slight degree, the suggestion 
which I have made. 


Delaget has lately made use of the principle of specific rate of growth, 
in considering the question of heredity itself. We know that the chromatin 
of the fertilised egg comes from the male and female parent alike, in equal or 
nearly equal shares; we know that the initial chromatin, so contributed, 
multiplies many thousand-fold, to supply the chromatin for every cell of the 
offspring’s body; and it has, therefore, a high “coefficient of growth.” If we 
admit, with Van Beneden and others, that the initial contributions of male and 
female chromatin continue to be transmitted to the succeeding generations 
of cells, we may then conceive these chromatins to retain each its own coefficient 
of growth; and if these differed ever so little, a gradyal preponderance of one 
or other would make itself felt in time, and might conceivably explain the 
preponderating influence of one parent or the other upon the characters of 
the offspring. Indeed O. Hertwig is said (according to Delage’s interpretation) 
to have actually shewn that we can artificially modify the rate of growth of 
one or other chromatin, and so increase or diminish the influence of the maternal 
or paternal heredity. This theory of Delage’s has its fascination, but it calls 
for somewhat large assumptions; and in particular, it seems (like so many 
other theories relating to the chromosomes) to rest far too much upon material 
elements, rather than on the imponderable dynamic factors of the cell. 


* J. Exp. Zool. vu, p. 457, 1909. 
} Biologica, ut, p. 161, June, 1913. 


154 THE RATE OF GROWTH [CH. 


We may summarise, as follows, the main results of the fore- 
going discussion : | 

(1) Except in certain minute organisms and minute parts of 
organisms, whose form is due to the direct action of molecular 
forces, we may look upon the form of the organism as a “function 
of growth,” or a direct expression of a rate of growth which varies 
according to its different directions. 

(2) Rate of growth is subject to definite laws, and the 
velocities in different directions tend to maintain a ratio which is 
more or less constant for each specific organism; and to this 
regularity is due the fact that the form of the organism is in general 
regular and constant. 

(3) Nevertheless, the ratio of velocities in different directions 
is not absolutely constant, but tends to alter or fluctuate in a 
regular way; and to these progressive changes are due the 
changes of form which accompany “development,” and the slower 
changes of form which continue perceptibly in after life. 

(4) The rate of growth is a function of the age of the organism , 
it has a maximum somewhat early in life, after which epoch of 
maximum it slowly declines. 

(5) The rate of growth is directly affected by temperatude, 
and by other physical conditions. 

(6) It is markedly affected, in the way of acceleration or 
retardation, at certain physiological epochs of life, such as birth, 
puberty, or metamorphosis. 

(7) Under certain circumstances, growth may be negative, the 
organism growing smaller: and such negative growth is a common 
accompaniment of metamorphosis, and a frequent accompaniment 
of old age. 

(8) The phenomenon of regeneration is associated with a large 
temporary increase in the rate of growth (or “acceleration” of 
growth) of the injured surface; in other respects, regenerative 
growth is similar to ordinary growth in all its essential phenomena. 


In this discussion of growth, we have left out of account a 
vast number of processes, or phenomena, by which, in the physio- 
logical mechanism of the body, growth is effected and controlled. 
We have dealt with growth in its relation to magnitude, and to 


111] CONCLUSION AND SUMMARY 155 


that relativity of magnitudes which constitutes form; and so we 
_ have studied it as a phenomenon which stands at the beginning 
of a morphological, rather than at the end of a physiological 
enquiry. Under these restrictions, we have treated it as far as 
possible, or in such fashion as our present knowledge permits, on 
strictly physical lines. 

In all its aspects, and not least in its relation to form, the 
growth of organisms has many analogies, some close and some 
perhaps more remote, among inanimate things. As the waves 
grow when the winds strive with the other forces which govern 
the movements of the surface of the sea, as the heap grows when 
we pour corn out of a sack, as the crystal grows when from the 
surrounding solution the proper molecules fall into their appro- 
priate places: so in all these cases, very much as in the organism 
itself, is growth accompanied by change of form, and by a develop- 
ment of definite shapes and contours. And in these cases (as 
in all other mechanical phenomena), we are led to equate our 
various magnitudes with time, and so to recognise that growth is 
essentially a question of rate, or of velocity. 

The differences of form, and changes of form, which are brought 
about by varying rates (or “laws’’) of growth, are essentially the 
same phenomenon whether they be, so to speak, episodes in the 
life-history of the individual, or manifest themselves as the normal 
and distinctive characteristics of what we call separate species of 
the race. From one form, or ratio of magnitude, to another there 
is but one straight and direct road of transformation, be the 
journey taken fast or slow; and if the transformation take place 
at all, it will in all likelihood proceed in the self-same way, whether 
it occur within the life-time of an individual or during the long 
ancestral history of a race. No small part of what is known as 
Wolfi’s or von Baer’s law, that the individual organism tends to 
pass through the phases characteristic of its ancestors, or that the 
life-history of the individual tends to recapitulate the ancestral 
history of its race, lies wrapped up in this simple account of the 
relation between rate of growth and form. 

But enough of this discussion. Let us leave for a while the 
subject of the growth of the organism, and attempt to study the 
conformation, within and without, of the individual cell. 


CHAPTER IV 


ON THE INTERNAL FORM AND STRUCTURE OF THE CELL 


In the early days of the cell-theory, more than seventy years 
ago, Goodsir was wont to speak of cells as “centres of growth” 
or “centres of nutrition,’ and to consider them as essentially 
“centres of force.” He looked forward to a time when the forces 
connected with the cell should be particularly investigated: when, 
that is to say, minute anatomy should be studied in its dynamical 
aspect. ““When this branch of enquiry,” he says “shall have 
been opened up, we shall expect to have a science of organic 
forces, having direct relation to anatomy, the science of organic 
forms*.” And likewise, long afterwards, Giard contemplated a 
science of morphodynamique,—but still looked upon it as forming 
so guarded and hidden a “territoire scientifique, que la plupart 
des naturalistes de nos jours ne le verront que comme Moise vit 
la terre promise, seulement de loin et sans pouvoir y entrery.” 

To the external forms of cells, and to the forces which produce 
and modify these forms, we shall pay attention in a later chapter. 
But there are forms and configurations of matter within the cell, 
which also deserve to be studied with due regard to the forces, 
known or unknown, of whose resultant they are the visible 
expression. . 

In the long interval since Goodsir’s day, the visible structure, 
the conformation and configuration, of the cell, has been studied 
far more abundantly than the purely dynamic problems that are 
associated therewith. The overwhelming progress of microscopic 
observation has multiplied our knowledge of cellular and intra- 
cellular structure; and to the multitude of visible structures it 

* Anatomical and Pathological Observations, p. 3, 1845; Anatomical Memoirs, 
II, p. 392, 1868. 


+ Giard, A., L’muf et les débuts de l’évolution, Bull. Sci. du Nord de la Fr. 
VIII, pp. 252-258, 1876. 


cH. Iv] INTERNAL FORM AND STRUCTURE OF CELL 157 


has been often easier to attribute virtues than to ascribe intelligible 
functions or modes of action. But here and there nevertheless, 
throughout the whole literature of the subject, we find recognition 
of the inevitable fact that dynamical problems lie behind the 
morphological problems of the cell. 

Biitschl pointed out forty years ago, with emphatic clearness, 
the failure of morphological methods, and the need for physical 
methods, if we were to penetrate deeper into the essential nature 
of the cell*. And such men as Loeb and Whitman, Driesch and 
Roux, and not a few besides, have pursued the same train of 
thought and similar methods of enquiry. 

Whitman f, for instance, puts the case in a nutshell when, in 
speaking of the so-called “caryokinetic” phenomena of nuclear 
division, he reminds us that the leading idea in the term “caryo- 
kinesis” is motvon,—‘“motion viewed as an exponent of forces 
residing in, or acting upon, the nucleus. It regards the nucleus 
as a seat of energy, which displays itself in phenomena of motion t.” 

In short 1t would seem evident that, except in relation to a 
dynamical investigation, the mere study of cell structure has but 
little value of its own. That a given cell, an ovum for instance, 
contains this or that visible substance or structure, germinal 
vesicle or germinal spot, chromatin or achromatin, chromosomes 
or centrosomes, obviously gives no explanation of the activities of 
the cell. And in all such hypotheses as that of “pangenesis,”’ in 
all the theories which attribute specific properties to micellae, 


* Entwickelungsvorgdnge der Hizelle, 1876; Investigations on Microscopic Foams 
and Protoplasm, p. 1, 1894. 

+ Journ. of Morphology, 1, p. 229, 1887. 

~ While it has been very common to look upon the phenomena of mitosis as- 
sufficiently explained by the results towards which they seem to lead, we may find 
here and there a strong protest against this mode of interpretation. The following 
is a case in point: “Ona tenté d’établir dans la mitose dite primitive plusieurs 
catégories, plusieurs types de mitose. On a choisi le plus souvent comme base 
de ces systémes des concepts abstraits et téléologiques: répartition plus ou moins 
exacte de la chromatine entre les deux noyaux-fils suivant qu’il y a ou non des 
chromosomes (Dangeard), distribution particuliere et signification dualiste des 
substances nucléaires (substance kinétique et substance générative ou héréditaire, 
Hartmann et ses éléves), etc. Pour moi tous ces essais sont a rejeter catégorique- 
ment a cause de leur caractére finaliste; de plus, ils sont construits sur des concepts 
non démontrés, et qui parfois représentent des généralisations absolument erronées.”’ 
A. Alexeieff, Archiv fiir Protistenkunde, x1x, p. 344, 1913. 


158 ON THE INTERNAL FORM AND [CH. 


idioplasts, ids, or other constituent particles of protoplasm or of 
the cell, we are apt to fall into the error of attributing to matter 
what is due to energy and is manifested in force: or, more strictly 
speaking, of attributing to material particles individually what is 
due to the energy of their collocation. 

The tendency is a very natural one, as knowledge of structure 
increases, to ascribe particular virtues to the material structures 
themselves, and the error is one into which the disciple is likely 
to fall, but of which we need not suspect the master-mind. The 
dynamical aspect of the case was in all probability kept well in 
view by those who, like Goodsir himself, first attacked the problem 
of the cell and originated our conceptions of its nature and 
functions. 

But if we speak, as Weismann and others speak, of an 
“hereditary substance,’ a substance which is split off from the 
parent-body, and which hands on to the new generation the 
characteristics of the old, we can only justify our mode of speech 
by the assumption that that particular portion of matter is the 
essential vehicle of a particular charge or distribution of energy, 
in which is involved the capability of producing motion, or of 
doing “work.” 

For, as Newton said, to tell us that a thing “is endowed with ~ 
an occult specific quality, by which it acts and produces manifest 
effects, is to tell us nothing; but to derive two or three general 
principles of motion* from phenomena would be a very great step 
in philosophy, though the causes of these principles were not yet 
discovered.’ The things which we see in the cell are less important 
than the actions which we recognise in the cell; and these latter 
we must especially scrutinize, in the hope of discovering how far 
they may be attributed to the simple and well-known physical 
forces, and how far they be relevant or irrelevant to the phenomena 
which we associate with, and deem essential to, the manifestation 
of life. It may be that in this way we shall in time draw nigh to 
the recognition of a specific and ultimate residuum. 


* This is the old philosophic axiom writ large: Ignorato motu, ignoratur 
natura; which again is but an adaptation of Aristotle’s phrase, 7 dpx} Tis Kwijoews, 
as equivalent to the “Efficient Cause.” FitzGerald holds that ‘all explanation 
consists in a description of underlying motions”; Scientific Writings, 1902, p. 385. 


Iv] STRUCTURE OF THE CELL 159 


And lacking, as we still do lack, direct knowledge of the actual 
forces inherent in the cell, we may yet learn something of their 
distribution, if not also of their nature, from the outward and 
inward configuration of the cell, and from the changes taking 
place in this configuration; that is to say from the movements 
of matter, the kinetic phenomena, which the forces in action set up. 

The fact that the germ-cell develops into a very complex 
structure, is no absolute proof that the cell itself is structurally 
avery complicated mechanism: nor yet, though this is somewhat 
less obvious, is it sufficient to prove that the forces at work, or 
latent, within it are especially numerous and complex. If we blow 
into a bowl of soapsuds and raise a great mass of many-hued and 
variously shaped bubbles, if we explode a rocket and watch the 
regular and beautiful configuration of its falling streamers, if we 
consider the wonders of a hmestone cavern which a filtering stream 
has filled with stalactites, we soon perceive that in all these cases 
we have begun with an initial system of very slight complexity, 
whose structure in no way foreshadowed the result, and whose 
comparatively simple intrinsic forces only play their part by 
complex interaction with the equally simple forces of the surround- 
ing medium. In an earlier age, men sought for the visible embryo, 
even for the homunculus, within the reproductive cells; and to 
this day, we scrutinize these cells for visible structure, unable to 
free ourselves from that old doctrine of “ pre-formation*.” 

Moreover, the microscope seemed to substantiate the idea 
{which we may trace back to Leibnizt and to Hobbest), that 
there is no limit to the mechanical complexity which we may 
postulate in an organism, and no limit, therefore, to the hypo- 
theses which we may rest thereon. 

But no microscopical examination of a stick of sealing-wax, 
no study of the material of which it is composed, can enlighten 


* As when Nageli concluded that the organism is, in a certain sense, “ vorge- 
bildet” ; Beitr. zur wiss. Botanik, u, 1860. Cf. E. B. Wilson, The Cell, etc., p. 302. 

+ “La matiere arrangée par une sagesse divine doit étre essentiellement organisée 
partout...il y a machine dans les parties de la machine Naturelle a Vinfini.”” Sur le 
principe de la Vie, p. 431 (Erdmann). ‘This is the very converse of the doctrine 
of the Atomists, who could not conceive a condition “ubi dimidiae partis pars 
semper habebit Dimidiam partem, nec res praefiniet ulla.” 

i Cf. an interesting passage from the Hlements (1, p. 445, Molesworth’s edit.), 
quoted by Owen, Hunterian Lectures on the Invertebrates, 2nd ed. pp. 40, 41, 1855. 


160 ON THE INTERNAL FORM AND [CH. 


us as to its electrical manifestations or properties. Matter of 
itself has no power to do, to make, or to become: it is in energy 
that all these potentialities reside, energy invisibly associated with 
the material system, and in interaction with the energies of the 
surrounding universe. 

That “function presupposes structure” has been declared an 
accepted axiom of biology. Who it was that so formulated the 
aphorism I do not know; but as regards the structure of the cell 
it harks back to Briicke, with whose demand for a mechanism, 
or organisation, within the cell histologists have ever since 
been attempting to comply*. But unless we mean to include 
thereby invisible, and merely chemical or molecular, structure, 
we come at once on dangerous ground. For we have seen, in 
a former chapter, that some minute “organisms” are already 
known of such all but infinitesimal magnitudes that everything 
which the morphologist is accustomed to conceive as “structure” 
has become physically impossible; and moreover recent research 
tends generally to reduce, rather than to extend, our conceptions 
of the visible structure necessarily inherent in living protoplasm. 
The microscopic structure which, in the last resort or in the simplest 
cases, it seems to shew, is that of a more or less viscous colloid, 
or rather mixture of colloids, and nothing more. Now, as Clerk 
Maxwell puts it, in discussing this very problem, “one material 
system can differ from another only in the configuration and 
motion which it has at a given instant 7.” If we cannot assume 
differences in structure, we must assume differences in motion, that 
is to say, in energy. And if we cannot do this, then indeed we are 
thrown back upon modes of reasoning unauthorised in physical 
science, and shall find ourselves constrained to assume, or to 
“admit, that the properties of a germ are not those of a purely 
material system.” 


* “Wir miissen deshalb den lebenden Zellen, abgesehen von der Molekular- 
structur der organischen Verbindungen welche sie enthalt, noch eine andere und 
in anderer Weise complicirte Structur zuschreiben, und diese es ist welche wir 
mit dem Namen Organisation bezeichnen,” Briicke, Die Elementarorganismen, 
Wiener Sitzungsber. xuiv, 1861, p. 386; quoted by Wilson, The Cell, etc. p. 289. 
Cf. also Hardy, Journ. of Physiol. xxtv, 1899, p. 159. 

t Precisely as in the Lucretian concursus, motus, ordo, positura, figurae, whereby 
bodies mutato ordine mutant naturam. 


Iv| STRUCTURE OF THE CELL 161 


But we are by no means necessarily in this dilemma. For 
though we come perilously near to it when we contemplate the 
lowest orders of magnitude to which life has been attributed, yet 
in the case of the ordinary cell, or ordinary egg or germ which is 
going to develop into a complex organism, if we have no reason 
to assume or to believe that it comprises an intricate “mechanism,” 
we may be quite sure, both on direct and indirect evidence, that, 
hke the powder in our rocket, it is very heterogeneous in its 
structure. It is a mixture of substances of various kinds, more 
or less fluid, more or less mobile, influenced in various ways by 
chemical, electrical, osmotic, and other forces, and in their 
admixture separated by a multitude of surfaces, or boundaries, at 
which these, or certain of these forces are made manifest. 

Indeed, such an arrangement as this is already enough to 
constitute a “mechanism”; for we must be very careful not to 
let our physical or physiological concept of mechanism be narrowed 
to an interpretation of the term derived from the delicate and 
complicated contrivances of human skill. From the physical 
point of view, we understand by a “mechanism” whatsoever 
checks or controls, and guides into determinate paths, the workings 
of energy; in other words, whatsoever leads in the degradation 
of energy to its manifestation in some determinate form of work, 
at a stage short of that ultimate degradation which lapses in 
uniformly diffused heat. This, as Warburg has well explained, is 
the general effect or function of the physiological machine, and in 
particular of that part of it which we call “cell-structure *.” 
The normal muscle-cell is something which turns energy, derived 
from oxidation, into work; it is a mechanism which arrests and 
utilises the chemical energy of oxidation in its downward course; 
but the same cell when injured or disintegrated, loses its “use- 
fulness,” and sets free a greatly increased proportion of its energy 
in the form of heat. 

But very great and wonderful things are done after this manner 
by means of a mechanism (whether natural or artificial) of 
extreme simplicity. A pool of water, by virtue of its suriace, 


* Otto Warburg, Beitrige zur Physiologie der Zelle, insbesondere iiber die 
Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s Ergebnisse der Physiologie, 
XIV, pp. 253-337, 1914 (see p. 315). (Cf. Bayliss, General Physiology, 1915, p. 590). 


its (Oe 11 


162 ON THE INTERNAL FORM AND [CH. 


is an admirable mechanism for the making of waves; with a lump 
of ice in it, it becomes an efficient and self-contained mechanism 
for the making of currents. The great cosmic mechanisms are 
stupendous in their simplicity; and, in point of fact, every great 
or little aggregate of heterogeneous matter (not identical in 
phase”) involves, zpso facto, the essentials of a mechanism. 
Even a non-living colloid, from its intrinsic heterogeneity, is in 
this sense a mechanism, and one in which energy is manifested 
in the movement and ceaseless rearrangement of the constituent 
particles. For this reason Graham (if I remember nightly) speaks 
somewhere or other of the colloid state as “the dynamic state of 
matter”; or in the same philosopher’s phrase (of which Mr 
Hardy* has lately reminded us), it possesses “energiat.” 

Let us turn then to consider, briefly and diagrammatically, the 
structure of the cell, a fertilised germ-cell or ovum for instance, 
not in any vain attempt to correlate this structure with the 
structure or properties of the resulting and yet distant organism ; 
but merely to see how far, by the study of its form and its changing 
internal configuration, we may throw light on certain forces which 
are for the time being at work within it. 

We may say at once that we can scarcely hope to learn more 
of these forces, in the first instance, than a few facts regarding 
their direction and magnitude; the nature and specific identity 
of the force or forces is a very different matter. This latter 
problem is likely to be very difficult of elucidation, for the reason, 
among others, that very different forces are often very much alike 
in their outward and visible manifestations. So it has come to 
pass that we have a multitude of discordant hypotheses as to the 
nature of the forces acting within the cell, and producing, in cell 
division, the “‘caryokinetic” figures of which we are about to 
speak. One student may, like Rhumbler, choose to account for 
them by an hypothesis of mechanical traction, acting on a reticular 
web of protoplasmi{; another, like Leduc, may shew us how in 

* Hardy, W. B., On some Problems of Living Matter (Guthrie Lecture), 
Tr. Physical Soc. London, xxviii, p. 99-118, 1916. 

+ As a matter of fact both phrases occur, side by side, in Graham’s classical 
paper on “Liquid Diffusion applied to Analysis,” Phil. Trans. cui, p. 184, 1861; 


Chem. and Phys. Researches (ed. Angus Smith), 1876, p. 554. 
{ L. Rhumbler, Mechanische Erklarung der Aehnlichkeit zwischen Magne-~ 


Iv] STRUCTURE OF THE CELL 163 


many of their most striking features they may be admirably 
simulated by the diffusion of salts in a colloid medium; others 
again, like Gallardo* and Hartog, and Rhumbler (in his earlier 
papers) t, insist on their resemblance to the phenomena of 
electricity and magnetism {; while Hartog believes that the force 
in question is only analogous to these, and has a specific identity 
of its own§. All these conflicting views are of secondary import- 
ance, so long as we seek only to account for certain configurations 
which reveal the direction, rather than the nature, of a force. 
One and the same system of lines of force may appear in a field 
of magnetic or of electrical energy, of the osmotic energy of 
diffusion, of the gravitational energy of a flowing stream. In short, 
we may expect to learn something of the pure or abstract dynamics, 
long before we can deal with the special physics of the cell. For 
indeed (as Maillard has suggested), just as uniform expansion 
about a single centre, to whatsoever physical cause it may be due 
will lead to the configuration of a sphere, so will any two centres 
or foci of potential (of whatsoever kind) lead to the configurations 
with which Faraday made us familiar under the name of “lines 
of force||”’; and this is as much as to say that the phenomenon, 


tischen Kraftliniensystemen und Zelltheilungsfiguren, Arch. f. Entw. Mech. xv, 
p- 482, 1903. 

* Gallardo, A., Essai d’interprétation des figures caryocinétiques, Anales del 
Museo de Buenos-Aires (2), 11, 1896; La division de la cellule, phénomene bipolaire 
de caractére électro-colloidal, Arch. f. Entw. Mech. xxvii, 1909, etc. 

+ Arch. f. Entw. Mech. m1, tv, 1896-97. 

{ On various theories of the mechanism of mitosis, see (e.g.) Wilson, The Cell 
in Development, etc., pp. 100-114; Meves, Zelltheilung, in Merkel u. Bonnet’s 
Ergebnisse der Anatomie, etc., vil, vill, 1897-8; Ida H. Hyde, Amer. Journ. 
of Physiol. x11, pp. 241-275, 1905; and especially Prenant, A., Théories et inter- 
prétations physiques de la mitose, J. de l’ Anat. et Physiol. xuv1, pp. 511-578, 1910. 

§ Hartog, M., Une force nouvelle: le mitokinétisme, C.R. 11 Juli, 1910; 
Mitokinetism in the Mitotic Spindle and in the Polyasters, Arch. f. Entw. Mech. 
XXvu, pp. 141-145, 1909; cf. ibid. xu, pp. 33-64, 1914. Cf. also Hartog’s papers 
in Proc. R. S. (B), Lxxvi, 1905; Science Progress (n.s.), 1, 1907; Riv. di Scienza, 
um, 1908; C. R. Assoc. fr. pour 1 Avancem. des Sc. 1914, ete. 

|| The configurations, as obtained by the usual experimental methods, were 
of course known long before Faraday’s day, and constituted the “convergent and 
divergent magnetic curves” of eighteenth century mathematicians. As Leslie 
said, in 1821, they were ‘“‘regarded with wonder by a certain class of dreaming 
philosophers, who did not hesitate to consider them as the actual traces of an 
invisible fluid, perpetually circulating between the poles of the magnet.” Faraday’s 
great advance was to interpret them as indications of stress in a mediwm,—of 


1]—2 


164 ON THE INTERNAL FORM AND [CH. 


though physical in the concrete, is in the abstract purely mathe- 
matical, and in its very essence is neither more nor less than a 
property of three-dimensional space. 

But as a matter of fact,-in this instance, that is to say in 
trying to explain the leading phenomena of the caryokinetic 
division of the cell, we shall soon perceive that any explanation 
which is based, like Rhumbler’s, on mere mechanical traction, is 
obviously inadequate, and we shall find ourselves limited to the 
hypothesis of some polarised and polarising force, such as we deal — 
with, for instance, in the phenomena of magnetism or electricity. 

Let us speak first of the cell itself, as it appears in a state of 
rest, and let us proceed afterwards to study the more active 
phenomena which accompany its division. 


Our typical cell is a spherical body; that is to say, the uniform 
surface-tension at its boundary is balanced by the outward 
resistance of uniform forces within. But at times the surface- 
tension may be a fluctuating quantity, as when it produces the 
rhythmical contractions or ““Ransom’s waves” on the surface of 
a trout’s egg; or again, while the egg is in contact with other 
bodies, the surface-tension may be locally unequal and variable, 
giving rise to an amoeboid figure, as in the egg of Hydra*. 

Within the ovum is a nucleus or germinal vesicle, also spherical, 
and consisting as a rule of portions of “chromatin,” aggregated 
together within a more fluid drop. The fact has often been 
commented upon that, in cells generally, there is no correlation 
of form (though there apparently is of s¢ze) between the nucleus 
and the “‘cytoplasm,” or main body of the cell. So Whitmany 
remarks that “except during the process of division the nucleus 
seldom departs from its typical spherical form. It divides and 
sub-divides, ever returning to the same round or oval form.... 
How different with the cell. It preserves the spherical form as 
rarely as the nucleus departs from it. Variation in form marks 
the beginning and the end of every important chapter in its 


tension or attraction along the lines, and of repulsion transverse to the lines, of the 
diagram. 

* Cf. also the curious phenomenon in a dividing egg described as ‘‘ spinning ” 
by Mrs G. F. Andrews, J. of Morph. xu, pp. 367-389, 1897. 

+ Whitman, J. of Morph. a, p. 40, 1889. 


tv | STRUCTURE OF THE CELL 165 


history.” On simple dynamical grounds, the contrast is easily 
explained. So long as the fluid substance of the nucleus is quali- 
tatively different from, and incapable of mixing with, the fluid 
or semi-fluid protoplasm which surrounds it, we shall expect it 
to be, as it almost always is, of spherical form. For, on the one 
hand, it is bounded by a liquid film, whose surface-tension 1s 
uniform; and on the other, it is immersed in a medium which 
transmits on all sides a uniform fluid pressure *. For a similar 
reason the contractile vacuole of a Protozoon is spherical in form: 
it is just a “drop” of fluid, bounded by a uniform surface- 
tension and through whose boundary-film diffusion is taking place. 
But here, owing to the small difference between the fluid constitut- 
ing, and that surrounding, the drop, the surface-tension equi- 
librium is unstable; it is apt to vanish, and the rounded outline 
of the drop, like a burst bubble, disappears in a moment fT. 
The case of the spherical nucleus is closely akin to the spherical 
form of the yolk within the bird’s egg t. But if the substance of 
the cell acquire a greater solidity, as for instance in a muscle 


* “Souvent il n’y a qu'une séparation physique entre le cytoplasme et le suc 
nucléaire, comme entre deux liquides immiscibles, etc.;”? Alexeieff, Sur la mitose 
dite “primitive,” Arch. f. Protistenk. xx1x, p. 357, 1913. 

+ The appearance of “‘vacuolation” is a result of endosmosis or the diffusion 
of a less dense fluid into the denser plasma of the cell. Caeteris paribus, it is less 
apparent in marine organisms than in those of freshwater, and in many or most 
marine Ciliates and even Rhizopods a contractile vacuole has not been observed 
(Biitschli, in Bronn’s Protozoa, p. 1414); it is also absent, and probably for the same 
reason, in parasitic Protozoa, such as the Gregarines and the Entamoebae. Rossbach 
shewed that the contractile vacuole of ordinary freshwater Ciliates was very greatly 
diminished in a 5 per cent. solution of NaCl, and all but disappeared in a 1 per cent. 
solution of sugar (A7b. z. z. Inst. Wiirzburg, 1872, cf. Massart, Arch. de Biol. Lx, 
p- 515, 1889). Actinophrys sol, when gradually acclimatised to sea-water, loses its 
vacuoles, and vice versa (Gruber, Biol. Centralbl. tx, p. 22, 1889); and the same is 
true of Amoeba (Zuelzer, Arch. f. Entw. Mech. 1910, p. 632). The gradual enlarge- 
ment of the contractile vacuole is precisely analogous to the change of size of a 
bubble until the gases on either side of the film are equally diffused, as described 
long ago by Draper (Phil. Mag. (n.s.), x1, p. 559, 1837). Rhumbler has shewn 
that contractile or pulsating vacuoles may be well imitated in chloroform-drops, 
suspended in water in which various substances are dissolved (Arch. f. Entw. 
Mech. vu, 1898, p. 103). The pressure within the contractile vacuole. always 
greater than without, diminishes with its size, being inversely proportional to 
its radius; and when it lies near the surface of the cell, as in a Heliozoon, it 
bursts as soon as it reaches a thinness which its viscosity or molecular cohesion no 
longer permits it to maintain. 

t Cf. p. 660. 


166 ON THE INTERNAL FORM AND [CH. 


cell, or by reason of mucous accumulations in an epithelium cell, 
then the laws of fluid pressure no longer apply, the external 
pressure on the nucleus tends to become unsymmetrical, and its 
shape is modified accordingly. “‘Amoeboid’”? movements may be 
set up in the nucleus by anything which disturbs the symmetry of 
its own surface-tension. And the cases, as in many Rhizopods, 
where “‘nuclear material” is scattered in small portions throughout 
the cell instead of being aggregated in a single nucleus, are probably 
capable of very simple explanation by supposing that the “phase 
difference” (as the chemists say) between the nuclear and the 
protoplasmic substance is comparatively shght, and the surface- 
tension which tends to keep them separate is correspondingly 
small *. 

It has been shewn that ordinary nuclei, isolated in a living 
or fresh state, easily flow together; and this fact is enough to 
suggest that they are aggregations of a particular substance rather 
than bodies deserving the name of particular organs. It is by 
reason of the same tendency to confluence or aggregation of 
particles that the ordinary nucleus is itself formed, until the 
imposition of a new force leads to its disruption. 

Apart from that invisible or ultra-microscopic heterogeneity 
which is inseparable from our notion of a “colloid,” there is a 
visible heterogeneity of structure within both the nucleus and the 
outer protoplasm. The former, for instance, contains a rounded 
nucleolus or “germinal spot,” certain conspicuous granules or 
strands of the peculiar substance called chromatin, and a coarse 
meshwork of a protoplasmic material known as “linin” or achro- 
matin; the outer protoplasm, or cytoplasm, is generally believed 
to consist throughout of a sponge-work, or rather alveolar mesh- 
work, of more and less fluid substances; and lastly, there are 
generally to be detected one or more very minute bodies, usually 
in the cytoplasm, sometimes within the nucleus, known as the 
centrosome or centrosomes. 

The morphologist is accustomed to speak of a 


ce 


polarity” of 


* The elongated or curved ‘‘macronucleus” of an Infusorian is to be looked 
upon as a single mass of chromatin, rather than as an aggregation of particles in 
a fluid drop, as in the case described. It has a shape of its own, in which ordinary 
surface-tension plays a very subordinate part. 


Iv] STRUCTURE OF THE CELL 167 


the cell, meaning thereby a symmetry of visible structure about 
a particular axis. For instance, whenever we can recognise in 
a cell both a nucleus and a centrosome, we may consider a 
line drawn through the two as the morphological axis of polarity : 
in an epithelium cell, it is obvious that the cell is morphologically 
symmetrical about a median axis passing from its free surface to 
its attached base. Again, by an extension of the term “polarity,” 
as is customary in dynamics, we may have a “radial” polarity, 
between centre and periphery; and lastly, we may have several 
apparently independent centres of polarity within the single cell. 
Only in cells of quite irregular, or amoeboid form, do we fail to 
recognise a definite and symmetrical “polarity.” The morpho- 
logical “polarity” is accompanied by, and is but the outward 
expression (or part of it) of a true dynamical polarity, or distribution 
of forces; and the “lines of force” are rendered visible by con- 
catenation of particles of matter, such as come under the influence 
of the forces in action. 

When the lines of force stream inwards from the periphery 
towards a point in the interior of the cell, the particles susceptible 
of attraction either crowd towards the surface of the cell, or, when 
retarded by friction, are seen forming lines or “‘fibrillae’’ which 
radiate outwards from the centre and constitute a so-called 
“aster.” In the cells of columnar or ciliated epithelium, where 
the sides of the cell are symmetrically disposed to their neighbours 
but the free and attached surfaces are very diverse from one 
another in their external relations, it is these latter surfaces which 
constitute the opposite poles; and in accordance with the parallel 
lines of force so set up, we very frequently see parallel lines of 
granules which have ranged themselves perpendicularly to the 
free surface of the cell (cf. fig. 97). 

A simple manifestation of “polarity” may be well illustrated 
by the phenomenon of diffusion, where we may conceive, and may 
automatically reproduce, a “field of force,” with its poles and 
visible lines of equipotential, very much as in Faraday’s conception 
of the field of force of a magnetic system. Thus, in one of Leduc’s 
experiments*, if we spread a layer of salt solution over a level 


* Théorie physico-chimique de la Vie, p. 73, 1910; Mechanism of Life, p. 56, 
1911. 


168 ON THE INTERNAL FORM AND [CH. 


plate of glass, and let fall into the middle of it a drop of indian 
ink, or of blood, we shall find the coloured particles travelling 
outwards from the central “pole of concentration” along the lines 
of diffusive force, and so mapping out for us a “monopolar field” 
of diffusion: and if we set two such drops side by side, their 
lines of diffusion will oppose, and repel, one another. Or, instead 
of the uniform layer of salt solution, we may place at a little 
distance from one another a grain of salt and a drop of blood, 
representing two opposite poles: and so obtain a picture of a 
“bipolar field” of diffusion. In either case, we obtain results 
closely analogous to the “morphological,” but really dynamical, 
polarity of the organic cell. But in all probability, the dynamical 
polarity, or asymmetry of the cell is a very complicated phenome- 
non: for the obvious reason that, in any system, one asymmetry 
will tend to beget another. A chemical asymmetry will induce an 
inequality of surface-tension, which will lead directly to a modifi- 
cation of form; the chemical asymmetry may in turn be due to a 
process of electrolysis in a polarised electrical field; and again 
the chemical heterogeneity may be intensified into a chemical 
“polarity,” by the tendency of certain substances to seek a locus 
of greater or less surface-energy. We need not attempt to 
grapple with a subject so complicated, and leading to so many 
problems which lie beyond the sphere of interest of the morph- 
ologist. But yet the morphologist, in his study of the cell, 
cannot quite evade these important issues; and we shall return 
to them again when we have dealt somewhat with the form of 
the cell, and have taken account of some of the simpler pheno- 
mena of surface-tension. 


We are now ready, and in some measure prepared, to study 
the numerous and complex phenomena which usually accompany 
the division of the cell, for instance of the fertilised egg. 

Division of the cell is essentially accompanied, and preceded, 
by a change from radial or monopolar to a definitely bipolar 
polarity. 

In the hitherto quiescent, or apparently quiescent cell, we per- 
ceive certain movements, which correspond precisely to what must 
accompany and result from a “polarisation” of forces within the 


Iv] STRUCTURE OF THE CELL 169 


cell: of forces which, whatever may be their specific nature, at least 
are capable of polarisation, and of producing consequent attraction 
or repulsion between charged particles of matter. The opposing 
forces which were distributed in equilibrium throughout the sub- 
stance of the cell become focussed at two “centrosomes,” which 
may or may not be already distinguished as visible portions of 
matter; in the egg, one of these is always near to, and the other 
remote from, the “animal pole” of the egg, which pole is visibly 
as well as chemically different from the other, and is the region in 
which the more rapid and conspicuous developmental changes will 
presently begin. Between the two centrosomes, a spindle-shaped 


Fig. 41. Caryokinetic figure in a dividing cell (or blastomere) of the Trout’s 
ege. (After Prenant, from a preparation by Prof. P. Bouin.) 


figure appears, whose striking resemblance to the lines of force 
made visible by iron-filings between the poles of a magnet, was at 
once recognised by Hermann Fol, when in 1873 he witnessed for 
the first time the phenomenon in question. On the farther side 
of the centrosomes are seen star-like figures, or “asters,” in which 
we can without difficulty recognise the broken lines of force which 
run externally to those stronger lines which lie nearer to the polar 
axis and which constitute the “spindle.” The lines of force are 
rendered visible or “material,” just as in the experiment of the 
iron-fil ngs, by the fact that, in the heterogeneous substance of 
. the cell, certain portions of matter are more “permeable” to the 
acting force than the rest, become themselves polarised after the 


170 ON THE INTERNAL FORM AND [CH. 


fashion of a magnetic or “ paramagnetic”’ body, arrange themselves 
in an orderly way between the two poles of the field of force, cling 
to one another as it were in threads*, and are only prevented by 
the friction of the surrounding medium from approaching and 
congregating around the adjacent poles. 

As the field of force strengthens, the more will the lines of force 
be drawn in towards the interpolar axis, and the less evident will 
be those remoter lines which constitute the terminal, or extrapolar, 
asters: a clear space, free from materialised lines of force, may 
thus tend to be set up on either side of the spindle, the 
so-called “Biitschli space” of the histologistst. On the other 
hand, the lines of force constituting the spindle will be less con- 
centrated if they find a path of less resistance at the periphery 
of the cell: as happens, in our experiment of the iron-filings, when 
we encircle the field of force with an iron ring. On this principle, 
the differences observed between cells in which the spindle is well 
developed and the asters small, and others in which the spindle 
is weak and the asters enormously developed, can be easily 
explained by variations in the potential of the field, the large, 
conspicuous asters being probably correlated with a marked 
permeability of the surface of the cell. 

The visible field of force, though often called the “nuclear 
spindle,” is formed outside of, but usually near to, the nucleus. 
Let us look a little more closely into the structure of this body, 
and into the changes which it presently undergoes. 

Within its spherical outline (Fig. 42), it contains an “‘alveolar” 


* Whence the name “mitosis” (Greek uiros, a thread), applied first by Flemming 
to the whole phenomenon. Kollmann (Biol. Centralbl. 1, p. 107, 1882) called it 
divisio per fila, or divisio laqueis implicata. Many of the earlier students, such as 
Van Beneden (Rech. sur la maturation de lceuf, Arch. de Biol. tv, 1883), and 
Hermann (Zur Lehre y. d. Entstehung d. karyokinetischen Spindel, Arch. f. mikrosk. 
Anat. xxxvul, 1891) thought they recognised actual muscular threads, drawing 
the nuclear material asunder towards the respective foci or poles; and some such 
view was long maintained by other writers, Boveri, Heidenhain, Flemming, R. 
Hertwig, and many more. In fact, the existence of contractile threads, or the 
ascription to the spindle rather than to the poles or centrosomes of. the active 
forces concerned in nuclear division, formed the main tenet of all those who declined 
to go beyond the “contractile properties of protoplasm” for an explanation of the 
phenomenon. (Cf. also J. W. Jenkinson, Q. J. M. S. xuvut, p. 471, 1904.) 

+ Cf. Biitschli, O., Ueber die kiinstliche Nachahmung der karyokinetischen 
Figur, Verh. Med. Nat. Ver. Heidelberg, v, pp. 28—41 (1892), 1897. 


Iv] STRUCTURE OF THE CELL 171 


meshwork (often described, from its appearance in optical section, 
as a “reticulum’’), consisting of more solid substances, with more 
fluid matter filling up the interalveolar meshes. This phenomenon 
is nothing else than what we call in ordinary language, a “froth” 
or a “foam.” It is a surface-tension phenomenon, due to the 
interacting surface-tensions of two intermixed fluids, not very 
different in density, as they strive to separate. Of precisely the 
same kind (as Biitschli was the first to shew) are the minute alveolar 
networks which are to be discerned in the cytoplasm of the cell*, 
and which we now know to be not inherent in the nature of 


\attraction-sphere 


| , centrosomes 


_ Spindle 


SSS SS sb SS 


\ 


| (ee 
| Moral its | 
Vat boy 
Cay vy 
\ . 
ne ' } / / 
aS 4 / 
Sk cele ite ” Sy Ay e— 
' ‘nuclear membrane 
nucleolus 
Fig. 42. Fig. 43. 


protoplasm, or of living matter in general, but to be due to various 
causes, natural as well as artificial. The microscopic honeycomb 
structure of cast metal under various conditions of cooling, even 
on a grand scale the columnar structure of basaltic rock, is an 
example of the same surface-tension phenomenon. 


* Arrhenius, in describing a typical colloid precipitate, does so in terms that 
are very closely applicable to the ordinary microscopic appearance of the protoplasm 
of the cell. The precipitate consists, he says, ‘““en un réseau d’une substance 
solide contenant peu d’eau, dans les mailles duquel est inclus un fluide contenant 
un peu de colloide dans beaucoup d’eau...Evidemment cette structure se forme 
a cause de la petite différence de poids spécifique des deux phases, et de la con- 
sistance gluante des particules séparées, qui s’attachent en forme de réseau.” Rev. 
Scientifique, Feb. 1911. 


172 ON THE INTERNAL FORM AND [CH. 


But here we touch the brink of a subject so important that we must not 
pass it by without a word, and yet so contentious that we must not enter into 
its details. The question involved is simply whether the great mass of 
recorded observations and accepted beliefs with regard to the visible structure 
of protoplasm and of the cell constitute a fair picture of the actual living cell, 
or be based on appearances which are incident to death itself and to the 
artificial treatment which the microscopist is accustomed to apply. The great 
bulk of histological work is done by methods which involve the sudden killing 
of the cell or organism by strong reagents, the assumption being that death 
is so rapid that the visible phenomena exhibited during life are retained or 
“fixed” in our preparations. While this assumption is reasonable and 
justified as regards the general outward form of small organisms or of individual 
cells, enough has been done of late years to shew that the case is totally 
different in the case of the minute internal networks, granules, etc., which 
represent the alleged structure of protoplasm. For, as Hardy puts it, “It is 
notorious that the various fixing reagents are coagulants of organic colloids, 
and that they produce precipitates which have a certain figure or structure,... 
and that the figure varies, other things being equal, according to the reagent 
used.”” So it comes to pass that some writers* have altogether denied the 
existence in the living cell-protoplasm of a network or alveolar “‘foam” ; 
others} have cast doubts on the main tenets of recent histology regarding 
nuclear structure; and Hardy, discussing the structure of certain gland-cells, 
declares that “‘there is no evidence that the structure discoverable in the cell- 
substance of these cells after fixation has any counterpart in the cell when 
living.” “A large part of it” he goes on to say “is an artefact. The 
profound difference in the minute structure of a secretory cell of a mucous 
gland according to the reagent which is used to fix it would, it seems 
to me, almost suffice to establish this statement in the absence of other 
evidence.” 

Nevertheless, histological study proceeds, especially on the part of the 
morphologists, with but little change in theory or in method, in spite of these 
and many other warnings. That certain visible structures, nucleus, vacuoles, 
“attraction-spheres” or centrosomes, etc., are actually present in the living 
cell, we know for certain; and to this class belong the great majority of 
structures (including the nuclear “spindle” itself) with which we are at present 
concerned. That many other alleged structures are artificial has also been 
placed beyond a doubt; but where to draw the dividing line we often do not 
know fi. 


* F. Schwartz, in Cohn’s Beitr. z. Biologie der Pflanzen, v, p. 1, 1887. 

+ Fischer, Anat. Anzeiger, Ix, p. 678, 1894, x, p. 769, 1895. 

f See, in particular, W. B. Hardy, On the structure of Cell Protoplasm, Journ. 
of Physiol. xxiv, pp. 158-207, 1889; also Héber, Physikalische Chemie der Zelle 
und der Gewebe, 1902. Cf. (int. al.) Flemming, Zellsubstanz, Kern und Zelltheilung 
1882, p. 51, ete. : 


Iv] STRUCTURE OF THE CELL 17a 


The following is a brief epitome of the visible changes undergone 
by a typical cell, leading up to the act of segmentation, and con- 
stituting the phenomenon of mitosis or caryokinetic division. In 
the egg of a sea-urchin, we see with almost diagrammatic com- 
pleteness what is set forth here*. 

1. The chromatin, which to begin with was distributed in 
granules on the otherwise achromatic reticulum (Fig. 42), concen- 
trates to form a skein or-spireme, which may be a continuous 
thread from the first (Figs. 43, 44), or from the first segmented. 
In any case it divides transversely sooner or later into a number 
of chromosomes (Fig. 45), which as a rule have the shape of little 


Spireme 


wt 


Z eS 


i 
\ \ 
“a / . 
\ mS Yl / S \, 
Hf 
By 


(Si 


chromosomes 


Fig. 44. Fig. 45. 


rods, straight or curved, often bent into a V, but which may 
also be ovoid, or round, or even annular. Certain deeply staining 
masses, the nucleoli, which may be present in the resting nucleus, 
do not take part in the process of chromosome formation; they 
are either cast out of the nucleus and are dissolved in the cyto- 
plasm, or fade away 7 situ. 

2. Meanwhile, the deeply staining granule (here extra- 
nuclear), known as the centrosome, has divided in two. The two 
resulting granules travel to opposite poles of the nucleus, and 


* Mv description and diagrams (Figs 42—51) are based on those of Professor 
E. B. Wilson. 


174 ON THE INTERNAL FORM AND [CH. 


7 


there each becomes surrounded by a system of radiating lines, the 
asters; immediately around the centrosome is a clear space, the 
centrosphere (Figs. 43-45). Between the two centrosomes with 
their asters stretches a bundle of achromatic fibres, the spindle. 
3. The surface-film bounding the nucleus has broken down, 
the definite nuclear boundaries are lost, and the spindle now 
stretches through the nuclear material, in which he the chromo- 
somes (Figs. 45, 46). These chromosomes now arrange them- 
selves midway between the poles of the spindle, where they form 
what is called the equatorial plate (Fig. 47). 
_ 4. Each chromosome splits longitudinally into two: usually 


equatorial plate 
' \ Spindle fibres 
———! 


1 
ae = 
eke ' he , aster 
> - ' - 
~ 5 1 1 a. 
/ 1 tees 
s y 1 r \ 
' 1 
H \ 


\ " whe a 
~ ia eee 
a se Re 
SSS ad eee oe a= 


— 


Fig. 46. Fig. 47. 


at this stage,—but it is to be noticed that the splitting may have 
taken place so early as the spireme stage (Fig. 48). 
5. The halves of the split chromosomes now separate from 
one another, and travel in opposite directions towards the two 
poles (Fig. 49). As they move, it becomes apparent that the spindle 
consists of a median bundle of ‘‘fibres,” the central spindle, running 
from pole to pole, and a more superficial sheath of “mantle- 
fibres,’ to which the chromosomes seem to be attached, and by 
which they seem to be drawn towards the asters. 

6. The daughter chromosomes, arranged now in two groups, 
become closely crowded in a mass near the centre of each aster 


Iv] STRUCTURE OF THE CELL 175 


(Fig. 50). They fuse together and form once more an alveolar reti- 
culum and may occasionally at this stage form another spireme. 


central spindie 
pas a Tia oa ae mantle-fibres 
> ~ ae ™ 
a UR re 


ee de hos 


' ' a 
nuclaolus ;__. nucleolus 
‘split chromosomes 


Fig. 48. Fig. 49. 


A boundary or surface wall is now developed round each recon- 
structed nuclear mass, and the spindle-fibres disappear (Fig. 51). 
The centrosome remains, as a rule, outside the nucleus. 


\Asoppearing spindle 


Sea Ss 
we roa 
, x 
i ms \ 
\ 
\ / \ 
=e 2 ae : had \ 
== & faery | (ES 
Ha = = Sore \ SEY LA | 
\ . / 
. NR I ie 4 
Naa ear, 
"cell- plate Reconstructed daughter-nucles 
Fig. 50. Fig. 51, 


7. On the central spindle, in the position of the equatorial 
plate, there has appeared during the migration of the chromosomes, 
a “cell-plate” of deeply staining thickenings (Figs. 50, 51). This 
is more conspicuous in plant-cells. 


176 ON THE INTERNAL FORM AND [CH. 


8. A constriction has meanwhile appeared in the cytoplasm, 
and the cell divides through the equatorial plane. In plant-cells 
the line of this division is foreshadowed by the “cell-plate,” which 
extends from the spindle across the entire cell, and splits into 
two layers, between which appears the membrane by which the 
daughter cells are cleft asunder. In animal cells the cell-plate 
does not attain such dimensions, and no cell-wall is formed. 


The whole, or very nearly the whole of these nuclear phenomena 
may be brought into relation with that polarisation of forces, in 
the cell as a whole, whose field is made manifest by the “spindle” 
and “asters” of which we have already spoken: certain particular 
phenomena, directly attributable to surface-tension and diffusion, 
taking place in more or less obvious and inevitable dependence 
upon the polar system *. 

At the same time, in attempting to explain the phenomena, we 
cannot say too clearly, or too often, that all that we are meanwhile 
justified in doing is to try to shew that such and such actions he 
within the range of known physical actions and phenomena, or that 
known physical phenomena produce effects similar to them. We 
want to feel sure that the whole phenomenon is not sw generis, but 
is somehow or other capable of being referred to dynamical laws, 
and to the general principles of physical science. But when we 
speak of some particular force or mode of action, using it as an 
illustrative hypothesis, we must stop far short of the implication 
that this or that force is necessarily the very one which is actually 
at work within the living cell; and certainly we need not attempt 
the formidable task of trying to reconcile, or to choose between, 
the various hypotheses which have already been enunciated, or 
the several assumptions on which they depend. 


Any region of space within which action is manifested is a 
field of force; and a simple example is a bipolar field, in which 
the action is symmetrical with reference to the line joining two 
points, or poles, and also with reference to the “ equatorial” 
plane equidistant from both. We have such a “field of force” in 


* The reference numbers in the following account refer to the paragraphs and 
figures of the preceding summary of visible nuclear phenomena. 


! 


—* |. 


Iv | STRUCTURE OF THE CELL 177 


the neighbourhood of the centrosome of the ripe cell or ovum, 
when it is about to divide: and by the time the centrosome has 
divided, the field is definitely a bipolar one. 

The quality of a medium filling the field of force may be uniform, 
or it may vary from point to point. In particular, it may depend 
upon the magnitude of the field; and the quality of one medium 
may differ from that of another. Such variation of quality, 
within one medium, or from one medium to another, is capable 
of diagrammatic representation by a variation of the direction or 
the strength of the field (other conditions being the same) from the 
state manifested in some uniform medium taken as a standard. 
The medium is said to be permeable to the force, in greater or less 
degree than the standard medium, according as the variation of 
the density of the lines of force from the standard case, under 
otherwise identical conditions, is in excess or defect. A body 
placed im the medium will tend to move towards regions of greater or 
less force according as its permeability is greater or less than that of 


the surrounding medium*. In the common experiment of placing 


iron-filings between the two poles of a magnetic field, the filings 
have a very high permeability; and not only do they themselves 
become polarised so as to attract one another, but they tend to 
be attracted from the weaker to the stronger parts of the field, and 
as we have seen, were it not for friction or some other resistance, 
they would soon gather together around the nearest pole. But 
if we repeat the same experiment with such a metal as bismuth, 
which is very little permeable to the magnetic force, then the 
conditions are reversed, and the particles, being repelled from the 
stronger to the weaker parts of the field, tend to take up their 
position as far from the poles as possible. The particles have 
become polarised, but in a sense opposite to that of the surround- 
ing, or adjacent, field. 

Now, in the field of force whose opposite poles are eee by 


* If the word permeability be deemed too directly suggestive of the phenomena 
of magnetism we may replace it by the more general term of specific inductive 
capacity. This would cover the particular case, which is by no means an improbable 
one, of our phenomena being due to a “surface charge” borne by the nucleus 
itself and also by the chromosomes: this surface charge being in turn the result 
of a difference in inductive capacity between the body or particle and its surrounding 
medium. (Cf. footnote, p. 187.) 


Wie ele 12 


178 ON THE INTERNAL FORM AND [cH. 


the centrosomes the nucleus appears to act as a more or less perme- 
able body, as a body more permeable than the surrounding medium, 
that is to say the “cytoplasm” of the cell. It is accordingly 
attracted by, and drawn into, the field of force, and tries, as it 
were, to set itself between the poles and as far as possible from 
both of them. In other words, the centrosome-foci will be 
apparently drawn over its surface, until the nucleus as a whole 
is involved within the field of force, which is visibly marked out 
by the “‘spindle” (par. 3, Figs. 44, 45). 

If the field of force be electrical, or act in a fashion analogous » 
to an electrical field, the charged nucleus will have its surface- 
tensions diminished*: with the double result that the inner 
alveolar meshwork will be broken up (par. 1), and that the 
spherical boundary of the whole nucleus will disappear (par. 2). 
The break-up of the alveoli (by thinning and rupture of their 
partition walls) leads to the formation of a net, and the further 
break-up of the net may lead to the unravelling of a thread or 
“spireme” (Figs. 43, 44). 

Here there comes into play a fundamental principle which, — 
in so far as we require to understand it, can be explained in simple 
words. The effect (and we might even say the object) of drawing 
the more permeable body in between the poles, is to obtain an 
“easier path” by which the hnes of force may travel; but it is 
obvious that a longer route through the more permeable body 
may at length be found less advantageous than a shorter route 
through the less permeable medium. That is to say, the more 
permeable body will only tend to be drawn in to the field of force 
until a point is reached where (so to speak) the way round and 
the way through are equally advantageous. We should accordingly 
expect that (on our hypothesis) there would be found cases in 
which the nucleus was wholly, and others in which it was only 
partially, and in greater or less degree, drawn in to the field 
between the centrosomes. This is precisely what is found to 
occur in actual fact. Figs. 44 and 45 represent two so-called 
“types,” of a phase which follows that represented in Fig. 43. 
According to the usual descriptions (and in particular to Professor 


* On the effect of electrical influences in altering the surface-tensions of the 
colloid particles, see Bredig, Anorganische Fermente, pp. 15, 16, 1901. 


Iv] STRUCTURE OF THE CELL 179 


E. B. Wilson’s*), we are told that, in such a case as Fig. 44, the 
“primary spindle” disappears and the centrosomes diverge to 
opposite poles of the nucleus; such a condition bemg found in 
many plant-cells, and in the cleavage-stages of many eggs. In 
Fig. 45, on the other hand, the primary spindle persists, and 
subsequently comes to form the main or “central” spindle; 
-while at the same time we see the fading away of the nuclear 
membrane, the breaking up of the spireme into separate chromo- 
somes, and an ingrowth into the nuclear area of the “astral rays,” 
—all as in Fig. 46, which represents the next succeeding phase of 
Fig. 45. This condition, of Fig. 46, occurs in a variety of cases ; 
it is well seen in the epidermal cells of the salamander, and is 
also on the whole characteristic of the mode of formation of the 
“polar bodies.”’ It is clear and obvious that the two “types” 
correspond to mere differences of degree, and are such as would 
naturally be brought about by differences in the relative per- 
meabilities of the nuclear mass and of the surrounding cytoplasm, 
or even by differences in the magnitude of the former body. 

But now an important change takes place, or rather an 
important difference appears; for, whereas the nucleus as a whole 
tended to be drawn in to the stronger parts of the field, when it 
comes to break up we find, on the contrary, that its contained 
spireme-thread or separate chromosomes tend to be repelled to 
the weaker parts. Whatever this difference may be due to,— 
whether, for instance, to actual differences of permeability, or 
possibly to differences in “surface-charge,”’—the fact is that the 
chromatin substance now behaves after the fashion of a “dia- 
magnetic” body, and is repelled from the stronger to the weaker 
parts of the field. In other words, its particles, lying in the 
inter-polar field, tend to travel towards the equatorial plane 
thereof (Figs. 47, 48), and further tend to move outwards towards 
the periphery of that plane, towards what the histologist 
calls the ‘“‘mantle-fibres,” or outermost of the lines of force of 
which the spindle is made up (par. 5, Fig. 47). And if this com- 
paratively non-permeable chromatin substance come to consist of 
separate portions, more or less elongated in form, these portions, 
or separate “chromosomes,” will adjust themselves longitudinally, 


* The Cell, etc. p. 66. 
12—2 


180 ON THE INTERNAL FORM AND [CH. 


in a peripheral equatorial circle (Figs. 48, 49). This is precisely 
what actually takes place. Moreover, before the breaking up of 
the nucleus, long before the chromatin material has broken up 
into separate chromosomes, and at the very time when it is being 
fashioned into a “‘spireme,” this body already lies in a polar field, 
and must already have a tendency to set itself in the equatorial 
plane thereof. But the long, continuous spireme thread is unable, 
so long as the nucleus retains its spherical boundary wall, to 
adjust itself in a simple equatorial annulus; in striving to do so, 
it must tend to coil and “kink” itself, and in so doing (if all this 
be so), it must tend to assume the characteristic convolutions of 
the “‘spireme.” 

After the spireme has broken up into separate chromosomes, 
these particles come into a position of temporary, and unstable, 


ri A 


Fig. 52. Chromosomes, undergoing splitting and separation. 
(After Hatschek and Flemming, diagrammatised.) 


equilibrium near the periphery of the equatorial plane, and 
here they tend to place themselves in a symmetrical arrange- 
ment (Fig. 52). The particles are rounded, lnear, sometimes 
annular, similar in form and size to one another; and 
lying as they do in a fluid, and subject to a symmetrical system 
of forces, it is not surprising that they arrange themselves 
in a symmetrical manner, the precise arrangement depending 
on the form of the particles themselves. This symmetry may 
_ perhaps be due, as has already been suggested, to induced 
electrical charges. In discussing Brauer’s observations on the 
splitting of the chromatic filament, and the symmetrical arrange- 
ment of the separate granules, in Ascaris megalocephala, Lillie* 


* Lillie, R. S., Amer. J. of Physiol. vim, p. 282, 1903. 


Iv] . STRUCTURE OF THE CELL 181 


remarks: ‘‘This behaviour is strongly suggestive of the division 
of a colloidal particle under the influence of its surface electrical 
charge, and of the effects of mutual repulsion in keeping the 
products of division apart.” It is also probable.that surface- 
tensions between the particles and the surrounding protoplasm 
would bring about an identical result, and would sufficiently 
account for the obvious, and at first sight, very curious, symmetry. 
We know that if we float a couple of matches in water they tend 
to approach one another, till they lie close together, side by side; 
and, if we lay upon a smooth wet plate four matches, half broken 
across, a precisely similar attraction brings the four matches 
together in the fcrm of a symmetrical cross. Whether one of 
these, or some other, be the actual explanation of the phenomenon, 
it is at least plain that by some physical cause, some mutual and 


Fig. 53. Annular chromosomes, formed in the spermatogenesis of 
the Mole-cricket. (From Wilson, after Vom Rath.) 


symmetrical attraction or repulsion of the particles, we must seek 
to account for the curious symmetry of these so-called “ tetrads.”’ 
The remarkable annular chromosomes, shewn in Fig. 53, can also 
be easily imitated by means of loops of thread upon a soapy film 
when the film within the annulus is broken or its tension reduced. 


So far as we have now gone, there is no great difficulty in 
pointing to simple and familiar phenomena of a field of force 
which are similar, or comparable, to the phenomena which we 
witness within the cell. But among these latter phenomena 
there are others for which it is not so easy to suggest, in accordance 
with known laws, a simple mode of physical causation. It is not 
at once obvious how, in any simple system of symmetrical forces, 


182 ON THE INTERNAL FORM AND _[cH. 


the chromosomes, which had at first been apparently repelled 
from the poles towards the equatorial plane, should then be split 
asunder, and should presently be attracted in opposite directions, 
some to one pole and some to the other. Remembering that it is 
not our purpose to assert that some one particular mode of action 
is at work, but merely to shew that there do exist physical forces, 
or distributions of force, which are capable of producing the 
required result, I give the following suggestive hypothesis, which 
I owe to my colleague Professor W. Peddie. 

As we have’ begun by supposing that the nuclear, or chromo- 
somal matter differs in permeability from the medium, that is to 


say the cytoplasm, in which it les, let us now make the further 
assumption that its permeability is variable, and depends upon the 
strength of the field. 

In Fig. 54, we have a field of force (representing our cell), 
consisting of a homogeneous medium, and including two opposite 
poles: lines of force are indicated by full lines, and loci of constant 
magnitude of force are shewn by dotted lines. 

Let us now consider a body whose permeability (uw) depends 
on the strength of the field F. At two field-strengths, such as 
F,, F,, let the permeability of the body be equal to that of the 


Iv] STRUCTURE OF THE CELL 183 


medium, and let the curved line in Fig. 55 represent generally 
its permeability at other field-strengths; and let the outer and 
inner dotted curves in Fig. 54 represent respectively the loci of 
the field-strengths F, and F,. The body if it be placed in the 
medium within either branch of the inner curve, or outside the 
outer curve, will tend to move into the neighbourhood of the 
adjacent pole. If it be placed in the region intermediate to the 
two dotted curves, it will tend to move towards regions of weaker 
field-strength. 

The locus F, is therefore a locus of stable position, towards 
which the body tends to move; the locus F, is a locus of unstable 
position, from which it tends to move. If the body were placed 


‘\ 


Fb Fa 


across F,, it might be torn asunder into two portions, the split 
coinciding with the locus F,. 

Suppose a number of such bodies to be scattered throughout 
- the medium. Let at first the regions Ff, and F, be entirely outside 
the space where the bodies are situated: and, in making this 
supposition we may, if we please, suppose that the loci which we 
are calling F, and Ff, are meanwhile situated somewhat farther 
from the axis than in our figure, that (for instance) F, is situated 
where we have drawn F,,, and that F, is still further out. The 
bodies then tend towards the poles; but the tendency may be 
very small if, in Fig. 55, the curve and its intersecting straight line 
do not diverge very far from one another beyond F,; in other 


184 ON THE INTERNAL FORM AND [CH. 


words, if, when situated in this region, the permeability of the 
bodies is not very much in excess of that of the medium. 

Let the poles now tend to separate farther and farther from 
one another, the strength of each pole remaining unaltered; in 
other words, let the centrosome-foci recede from one another, as 
they actually do, drawing out the spindle-threads between them. 
The loci F,, F,,, will close in to nearer relative distances from the 
poles. In doing so, when the locus F, crosses one of the bodies, 
the body may be torn asunder; if the body be of elongated shape, 
and be crossed at more points than one, the forces at work will 
tend to exaggerate its foldings, and the tendency to rupture is 
greatest when F, is in some median position (Fig. 56). 

When the locus F, has passed entirely over the body, the body 
tends to move towards regions of weaker force; but when, in 


turn, the locus Ff, has crossed it, then the body again moves towards 
regions of stronger force, that.is to say, towards the nearest pole. 
And, in thus moving towards the pole, it will do so, as appears 
actually to be the case in the dividing cell, along the course of 
the outer lines of force, the so-called ‘“‘mantle-fibres” of the 
histologist*. 

Such considerations as these give general results, easily open 
to modification in detail by a change of any of the arbitrary 
postulates which have been made for the sake of simplicity. 
Doubtless there are many other assumptions which would more 
or less meet the case; for instance, that of Ida H. Hyde that, 


* We have not taken account in the above paragraphs of the obvious fact that 
the supposed symmetrical field of force is distorted by the presence in it of the 
more or less permeable bodies; nor is it necessary for us to do so, for to that 
distorted field the above argument continues to apply, word for word. 


Iv] STRUCTURE OF THE CELL 185 


during the active phase of the chromatin molecule (during which 
it decomposes and sets free nucleic acid) it carries a charge opposite 
- to that which it bears during its resting, or alkaline phase; and 
that it would accordingly move towards different poles under the 
influence of a current, wandering with its negative charge in an 
alkaline fluid during its acid phase to the anode, and to the kathode 
during its alkaline phase. A whole field of speculation is opened 
up when we begin to consider the cell not merely as a polarised 
electrical field, but also as an electrolytic field, full of wandering 
ions. Indeed it is high time we reminded ourselves that we have 
perhaps been dealing too much with ordinary physical analogies : 
and that our whole field of force within the cell is of an order of 
magnitude where these grosser analogies may fail to serve us, 
and might even play us false, or lead us astray. But our sole 
object meanwhile, as I have said more than once, is to demon- 
strate, by such illustrations as these, that, whatever be the actual 
and as yet unknown modus operandi, there are physical conditions 
and distributions of force which could produce just such phenomena 
of movement as we see taking place within the living cell. 
This, and no more, is precisely what Descartes is said to have 
claimed for his description of the human body as a “mechanism *.”’ 


The foregoing account is based on the provisional assumption 
that the phenomena of caryokinesis are analogous to, if not identical 
with those of a bipolar electrical field; and this comparison, in 
my opinion, offers without doubt the best available series of 
analogies. But we must on no account omit to.mention the 
fact that some of Leduc’s diffusion-experiments offer very remark- 
able analogies to the diagrammatic phenomena of caryokinesis, as 
shewn in the annexed figure+. Here we have two identical (not 
opposite) poles of osmotic concentration, formed by placing a drop 
of indian ink in salt water, and then on either side of this central 
_ drop, a hypertonic drop of salt solution more lightly coloured. 
On either side the pigment of the central drop has been drawn 
towards the focus nearest to it; but in the middle line, the pigment 


* M. Foster, Lectures on the History of Physiology, 1901, p. 62. 
+ Op. cit. pp. 110 and 91. 


186 ON THE INTERNAL FORM AND [on. 


is drawn in opposite directions by equal forces, and so tends to 
remain undisturbed, in the form of an “equatorial plate.” 

Nor should we omit to take account (however briefly and 
inadequately) of a novel and elegant hypothesis put forward by 
A.B. Lamb. This hypothesis makes use of a theorem of Bjerknes, _ 
to the effect that synchronously vibrating or pulsating bodies in 
a liquid field attract or repel one another according as their 
oscillations are identical or opposite in phase. Under such 
circumstances, true currents, or hydrodynamic lines of force, are” 
produced, identical in form with the lines of force of a magnetic 
field; and other particles floating, though not necessarily pulsating, 
in the liquid field, tend to be attracted or repelled by the pulsating 


Fig. 57. Artificial caryokinesis (after Leduc), for comparison 
with Fig. 41, p. 169. 


bodies according as they are lighter or heavier than the surrounding 
fluid. Moreover (and this is the most remarkable point of all), 
the lines of force set up by the oppositely pulsating bodies are the 
same as those which are produced by opposite magnetic poles: 
though in the former case repulsion, and in the latter case attrac- 
tion, takes place between the two poles*. 


But to return to our general discussion. 
While it can scarcely be too often repeated that our enquiry 
is not directed towards the solution of physiological problems, save 


* Lamb, A. B., A new Explanation of the Mechanism of Mitosis, Journ. Exp. 
Zool. Vv, pp. 27-33, 1908. 


Iv | STRUCTURE OF THE CELL 187 


only in so far as they are inseparable from the problems presented 
by the visible configurations of form and structure, and while we 
try, as far as possible, to evade the difficult question of what 
particular forces are at work when the mere visible forms produced 
are such as to leave this an open question, yet in this particular 
case we have been drawn into the use of electrical analogies, and 
we are bound to justify, if possible, our resort to this particular 
mode of physical action. There is animportant paper by R.S. Lillie, 
on the ‘‘ Electrical Convection of certain Free Cells and Nuclei*,” 
which, while I cannot quote it in direct support of the suggestions 
which I have made, yet gives just the evidence we need in order 
to shew that electrical forces act upon the constituents of the 
cell, and that their action discriminates between the two species 
of colloids represented by the cytoplasm and the nuclear chromatin. 
And the difference is such that, in the presence of an electrical 
current, the cell substance and the nuclei (including sperm-cells) 
tend to migrate, the former on the whole with the positive, the 
latter with the negative stream: a difference of electrical potential 
being thus indicated between the particle and the surrounding 
medium, just as in the case of minute suspended particles of various 
kinds in various feebly conducting media}. And the electrical 
difference is doubtless greatest, in the case of the cell constituents, 
just at the period of mitosis: when the chromatin is invariably 
in its most deeply staining, most strongly acid, and therefore, 
presumably, in its most electrically negative phase. In short, 


* Amer. J. of Physiol. vi, pp. 273-283, 1903 (vide supra, p. 181); cf. «bid. 
xv, pp. 46-84, 1905. Cf. also Biological Bulletin, tv, p. 175, 1903. 

+ In like manner Hardy has shewn that colloid particles migrate with the 
negative stream if the reaction of the surrounding fluid be alkaline, and vice versa. 
The whole subject is much wider than these brief allusions suggest, and is essentially 
part of Quincke’s theory of Electrical Diffusion or Endosmosis: according to 
which the particles and the fluid in which they float (or the fluid and the capillary 
walls through which it flows) each carry a charge, there being a discontinuity of 
potential at the surface of contact, and hence a field of force leading to powerful 
tangential or shearing stresses, communicating to the particles a velocity which 
varies with the density per unit area of the surface charge. See W. B. Hardy’s 
paper on Coagulation by Electricity, Journ. of Physiol. xx1v, p. 288-304, 1899, 
also Hardy and H. W. Harvey, Surface Electric Charges of Living Cells, Proc. R. S. 
LXxxtv (B), pp. 217-226, 1911, and papers quoted therein. Cf. also E. N. Harvey’s 
observations on the convection of unicellular organisms in an electric field (Studies 
on the Permeability of Cells, Journ. of Exper. Zool. x, pp. 508-556, 1911). 


188 ON THE INTERNAL FORM AND [CH. 


Lillie comes easily to the conclusion that “electrical theories of 
mitosis are entitled to more careful consideration than they have 
hitherto received.” 

Among other investigations, all leading towards the same 
general conclusion, namely that differences of electric potential 
play a great part in the phenomenon of cell division, I would 
mention a very noteworthy paper by Ida H. Hyde*, in which the 
writer shews (among other important observations) that not only — 
is there a measurable difference of potential between the animal 
and vegetative poles of a fertilised egg (Fundulus, toad, turtle, 
etc.), but that this difference is not constant, but fluctuates, or 
actually reverses its direction, periodically, at epochs coinciding 
with successive acts of segmentation or other important phases 
in the development of the eggt; just as other physical rhythms, 
for instance in the production of CO,, had already been shewn 
to do. Hence we shall be by no means surprised to find that the 
“materialised”’ lines of force, which in the earlier stages form the 
convergent curves of the spindle, are replaced in the later phases 
of caryokinesis by divergent curves, indicating that the two foci, 
which are marked out within the field by the divided and recon- 
stituted nuclei, are now alike in their polarity (Figs. 58, 59). 

It is certain, to my mind, that these observations of Miss 
Hyde's, and of Lillie’s, taken together with those of many writers 
on the behaviour of colloid particles generally in their relation 
to an electrical field, have a close bearing upon the physiological 
side of our problem, the full discussion of which hes outside our 
present field. 


The break-up of the nucleus, already referred to and ascribed 
to a diminution of its surface-tension, is accompanied by certain 
diffusion phenomena which are sometimes visible to the eye; and 
we are reminded of Lord Kelvin’s view that diffusion is implicitly 


* On Differences in Electrical Potential in Developing Eggs, Amer. Journ. of 
Physiol. xt1, pp. 241-275, 1905. This paper contains an excellent summary of 
various physical theories of the segmentation of the cell. 

+ Gray has recently demonstrated a temporary increase of electrical con- 
ductivity in sea-urchin eggs during the process of fertilisation (The Electrical 
Conductivity of fertilised and unfertilised Eggs, Journ. Mar. Biol. Assoc. x, pp. 
50-59, 1913). 


Iv | STRUCTURE OF THE CELL 189 


associated with surface-tension changes, of which the first step 
is a minute puckering of the surface-skin, a sort of interdigi- 
tation with the surrounding medium. For instance, Schewia- 
koff has observed in Huglypha* that, just before the break-up 
of the nucleus, a system of rays appears, concentred about it, 
but having nothing to do with the polar asters: and during the 
existence of this striation, the nucleus enlarges very considerably, 
evidently by imbibition of fluid from the surrounding protoplasm. 
In short, diffusion is at work, hand in hand with, and as it were 
in opposition to, the surface-tensions which define the nucleus. 


Fig. 58. Final stage in the first Fig. 59. Diagram of field of force 
segmentation of the egg of Cerebra- with two similar poles. 
tulus. (From Prenant, after Coe.) + 


By diffusion, hand in hand with surface-tension, the alveoli of 
the nuclear meshwork are formed, enlarged, and finally ruptyred: 
diffusion sets up the movements which give rise to the appearance 
of rays, or striae, around the nucleus: and through increasing 
diffusion, and weakening surface-tension, the rounded outline of 
the nucleus finally disappears. 

* Schewiakoff, Ueber die karyokinetische Kerntheilung der Huglypha alveolata, 
Morph. Jahrb. x1, pp. 193-258, 1888 (see p. 216). 


+ Coe, W. R., Maturation and Fertilization of the Egg of Cerebratulus, Zool. 
Jahrbiicher { Anat. Abth.), x1, pp. 425-476, 1899. 


190 ON THE INTERNAL FORM AND [CH. 


As we study these manifold phenomena, in the individual cases 
of particular plants and animals, we recognise a close identity of 
type, coupled with almost endless variation of specific detail; 
and in particular, the order of succession in which certain of the 
phenomena occur is variable and irregular. The precise order of 
the phenomena, the time of longitudinal and of transverse fission 
of the chromatin thread, of the break-up of the nuclear wall, and 
so forth, will depend upon various minor contingencies and 
“interferences.” And it is worthy of particular note that these 
variations, in the order of events and in other subordinate details, 
while doubtless attributable to specific physical conditions, would 
seem to be without any obvious classificatory value or other 
biological significance*. 


As regards the actual mechanical division of the cell into two 
halves, we shall see presently that, in certain cases, such as that 
of a long cylindrical filament, surface-tension, and what is known 
as the principle of “minimal area,” go a long way to explain the 
mechanical process of division; and in all cells whatsoever, the 
process of division must somehow be explained as the result of 
a conflict between surface-tension and its opposing forces. But 
in such a case as our spherical cell, it is not very easy to see what 
physical cause is at work to disturb its equilibrium and its integrity. 

The fact that, when actual division of the cell takes place, it 
does so at right angles to the polar axis and precisely in the 
direction of the equatorial plane, would lead us to suspect that 
the new surface formed in the equatorial plane sets up an annular 
tension, directed inwards, where it meets the outer surface layer 
of the cell itself. But at this point, the problem becomes more 
complicated. Before we could hope to comprehend it, we should 
have not only to enquire into the potential distribution at the 
surface of the cell in relation to that which we have seen to exist 
in its interior, but we should probably also have to take account 
of the differences of potential which the material arrangements 
along the lines of force must themselves tend to produce. Only 


* Thus, for example, Farmer and Digby (On Dimensions of Chromosomes 
considered in relation to Phylogeny, Phil. Trans. (B), ccv, pp. 1-23, 1914) have 
been at pains to shew, in confutation of Meek (ibid. com, pp. 1-74, 1912), that the 
width of the chromosomes cannot be correlated with the order of phylogeny. 


Iv| STRUCTURE OF THE CELL 191 


thus could we approach a comprehension of the balance of forces 
which cohesion, friction, capillarity and electrical distribution 
combine to set up. | 

The manner in which we regard the phenomenon would seem 
to turn, in great measure, upon whether or no we are justified in 
assuming that, in the liquid surface-film of a minute spherical cell, 
local, and symmetrically localised, differences of surface-tension 
are likely to occur. If not, then changes in the conformation of 
the cell such as lead immediately to its division must be ascribed 
not to local changes in its surface-tension, but rather to direct 
changes in internal pressure, or to mechanical forces due to an 
induced surface-distribution of electrical potential. 

It has seemed otherwise to many writers, and we have a number 
of theories of cell division which are all based directly on in- 
equalities or asymmetry of surface-tension. For instance, Biitschhi 
suggested, some forty years ago”, that cell division is brought 
about by an increase of surface-tension in the equatorial region 
of the cell. This explanation, however, can scarcely hold; for 
it would seem that such an increase of surface-tension in the 
equatorial plane would lead to the cell becoming flattened out into 
a disc, with a sharply curved equatorial edge, and to a streaming 
of material towards the equator. In 1895, Loeb shewed that the 
streaming went on from the equator towards the divided nuclei, 
and he supposed that the violence of these streaming movements 
brought about actual division of the cell: a hypothesis which was 
adopted by many other physiologists+. This streaming move- 
ment would suggest, as Robertson has pointed out, a diminution 
of surface-tengion in the region of the equator. Now Quincke has 
shewn that the formation of soaps at the surface of an oil-droplet 
results in a diminution of the surface-tension of the latter; and 
that if the saponification be local, that part of the surface tends to 
spread. By laying a thread moistened with a dilute solution of 
caustic alkali, or even merely smeared with soap, across a drop 
of oil, Robertson has further shewn that the drop at once divides 
into two: the edges of the drop, that is to say the ends of the 


* Cf. also Arch. f. Hntw. Mech. x, p. 52, 1900. 
+ Cf. Loeb, Am. J. of Physiol. v1, p. 432, 1902.; Erlanger, Biol. Centralbl. 
xvu, pp. 152, 339, 1897; Conklin, Biol. Lectures, Woods Holl, p. 69, etc. 1898-9. 


192 ON THE INTERNAL FORM AND [CH. 


diameter’ across which the thread les, recede from the thread, 
so forming a notch at each end of the diameter, while violent 
streaming motions are set up at the surface, away from the thread 
in the direction of the two opposite poles. Robertson* suggests, 
accordingly, that the division of the cell is actually brought about 
by a lowering of the equatorial surface-tension, and that this in 
turn is due to a chemical action, such as a liberation of cholin, 
or of soaps of cholin, through the splitting of lecithin in nuclear 
synthesis. 

But purely chemical changes are not of necessity the funda- 
mental cause of alteration in the surface-tension of the egg, for 
the action of electrolytes on surface-tension 1s now well known 
and easily demonstrated. So, according to other views than 
those with which we have been dealing, electrical charges are 
sufficient in themselves to account for alterations of surface- 
tension; while these in turn account for that protoplasmic 
streaming which, as so many investigators agree, initiates the 
segmentation of the eggf. A great part of our difficulty arises 
from the fact that in such a case as this the various phenomena 
are so entangled and apparently concurrent that it is hard to say 
which initiates another, and to which this or that secondary 
phenomenon may be considered due. Of recent years the pheno- 
menon of adsorption has been adduced (as we have already briefly 
said) in order to account for many of the events and appearances 
which are associated with the asymmetry, and lead towards the 
division, of the cell. But our short discussion of this phenomenon 
may be reserved for another chapter. 

However, we are not directly concerned here with the 
phenomena of segmentation or cell division in themselves, except 
only in so far as visible changes of form are capable of easy and 
obvious correlation with the play of force. The very fact of 
“development” indicates that, while it lasts, the equilibrium of 
the egg is never completet. And we may simply conclude the 

* Robertson, T. B., Note on the Chemical Mechanics of Cell Division, Arch. 
f. Entw. Mech. xxvu, p. 29, 1909, xxxv, p. 692, peu: Cf. R. S. Lillie, J. Hap. 
Zool. Xx, pp. 369—402, 1916. 

+ Cf. D’Arsonval, Arch. de Physiol. p. 460, 1889; Ida H. Hyde, op. cit. p. 242. 


t Cf. Plateau’s remarks (Statique des liquides, 1, p. 154) on the tendency towards: 
equilibrium, rather than actual equilibrium, in many of his systems of soap-films. 


Iv | STRUCTURE OF THE CELL 193 


matter by saying that, if you have caryokinetic figures developing 
inside the cell, that of itself indicates that the dynamic system 
and the localised forces arising from it are in continual alteration ; 
and, consequently, changes in the outward configuration of the 
system are bound to take place. 


As regards the phenomena of fertilisation,—of the union of 
the spermatozoon with the “pronucleus” of the egg,—we might 
study these also in illustration, up to a certain point, of the 
polarised forces which are manifestly at work. But we shall 
merely take, as a single illustration, the paths of the male and 
female pronuclei, as they travel to their ultimate meeting place. 

The spermatozoon, when within a very short distance of the 
egg-cell, is attracted by it. Of the nature of this attractive force 
we have no certain knowledge, though we would seem to have 
a pregnant hint in Loeb’s discovery that, in the neighbourhood 
of other substances, such even as a fragment, or bead, of glass, 
the spermatozoon undergoes a similar attraction. But, whatever 
the force may be, it is one acting normally to the surface of the 
ovum, and accordingly, after entry, the sperm-nucleus points 
straight towards the centre of the egg; from the fact that other 
spermatozoa, subsequent to the first, fail to effect an entry, we 
may safely conclude that an immediate consequence of the entry 
of the spermatozoon is an increase in the surface-tension of the 
ege*. Somewhere or other, near or far away, within the egg, lies 
its own nuclear body, the so-called female pronucleus, and we 
find after a while that this has fused with the head of the sperma- 
tozoon (or male pronucleus), and that the body resulting from 
their fusion has come to occupy the centre of the egg. This musi 
be due (as Whitman pointed out long ago) to a force of attraction 
acting between the two bodies, and another force acting upon 
one or other or both in the direction of the centre of the cell. 
Did we know the magnitude of these several forces, it would be 
a very easy task to calculate the precise path which the two 
pronuclei would follow, leading to conjugation and the central 


* But under artificial conditions, “‘polyspermy” may take place, e.g. under 


the action of dilute poisons, or of an abnormally high temperature, these being 
all, doubtless, conditions under which the surface-tension is diminished. 


La ee 113° 


194 ON THE INTERNAL FORM AND [CH. 


position. As we do not know the magnitude, but only the direction, 
of these forces we can only make a general statement: (1) the 
paths of both moving bodies will lie wholly within a plane triangle 
drawn between the two bodies and the centre of the cell; (2) unless 
the two bodies happen to lie, to begin with, precisely on a diameter 
of the cell, their paths until they meet one another will be curved 
paths, the convexity of the curve being towards the straight line 
joining the two bodies; (3) the two bodies will meet a little before 
they reach the centre; and, having met and fused, will travel 
on to reach the centre in a straight line. The actual study and 
observation of the path followed is not very easy, owing to the 
fact that what we usually see is not the path itself, but only a 
projection of the path upon the plane of the microscope; but the 
curved path is particularly well seen in the frog’s egg, where the 
path of the spermatozoon is marked by a little streak of brown 
pigment, and the fact of the meeting of the pronuclei before 
reaching the centre has been repeatedly seen by many observers. 

The problem is nothing else than a particular case of the 
famous problem of three bodies, which has so occupied the 
astronomers; and it is obvious that the foregoing brief description 
is very far from including all possible cases. Many of these are 
particularly described in the works of Fol, Roux, Whitman and 
others *. 


The intracellular phenomena of which we have now spoken 
have assumed immense importance in biological literature and 
discussion during the last forty years; but it is open to us to doubt — 
whether they will be found in the end to possess more than a 
remote and secondary biological significance. Most, if not all of 
them, would seem to follow immediately and inevitably from very 
simple assumptions as to the physical constitution of the cell, and 
from an extremely simple distribution of polarised forces within 
it. We have already seen that how a thing grows, and what it 
grows into, is a dynamic and not a merely material problem; so 
far as the material substance is concerned, it is so only by reason 


* Fol, H., Recherches sur la fécondation, 1879. Roux, W., Beitrige zur 
Entwickelungsmechanik des Embryo, Arch. f. Mikr. Anat. xrx, 1887. Whitman, 
C. O., Odkinesis, Journ. of Morph. 1, 1887. 


Iv | STRUCTURE OF THE CELL 195 


of the chemical, electrical or other forces which are associated 
with it. But there is another consideration which would lead us 
to suspect that many features in the structure and configuration 
of the cell are of very secondary biological importance; and that 
is, the great variation to which these phenomena are subject in 
similar or closely related organisms, and the apparent impossibility 
of correlating them with the peculiarities of the organism as a 
whole. ‘Comparative study has shewn that almost every detail 
of the processes (of mitosis) described above is subject to variation 
in different forms of cells*.” A multitude of cells divide to the 
accompaniment of caryokinetic phenomena; but others do so 
without any visible caryokinesis at all. Sometimes the polarised 
field of force is within, sometimes it is adjacent to, and at other 
times it lies remote from the nucleus. The distribution of potential 
is very often symmetrical and bipolar, as in the case described ; 
but a less symmetrical distribution often occurs, with the result that 
we have, for a time at least, numerous centres of force, instead 
of the two main correlated poles: this is the simple explanation 
of the numerous stellate figures, or “ Strahlungen,” which have 
been described in certain eggs, such as those of Chaetopterus. In 
one and the same species of worm (Ascaris megalocephala), one 
group or two groups of chromosomes may be present. And 
remarkably constant, in general, as the number of chromosomes in 
any one species undoubtedly is, yet we must not forget that, in 
plants and animals alike, the whole range of observed numbers is 
but a small one; for (as regards the germ-nuclei) few organisms 
have less than six chromosomes, and fewer still have more than 
sixteen}. In closely related animals, such as various species of 
Copepods, and even in the same species of worm or insect, the 
form of the chromosomes, and their arrangement in relation to 
the nuclear spindle, have been found to differ in the various ways 
alluded to above. In short, there seem to be strong grounds for 
believing that these and many similar phenomena are in no way 
specifically related to the particular organism in which they have 


* Wilson, Zhe Cell. p. 77. 

+ Eight and twelve are by much the commonest numbers, six and sixteen 
coming next in order. If we may judge by the list given by E. B. Wilson (The 
Cell, p. 206), over 80 % of the observed cases lie between 6 and 16, and nearly 
60 % between 8 and 12. 


13—2 


196 ON THE INTERNAL FORM AND [CH. 


been observed, and are not even specially and indisputably con- 
nected with the organism as such. They include such manifesta- 
tions of the physical forces, in their various permutations and 
combinations, as may also be witnessed, under appropriate 
conditions, in non-living things. 

When we attempt to separate our purely morphological or 
“purely embryological” studies from physiological and physical 
investigations, we tend zpso facto to regard each particular structure 
and configuration as an attribute, or a particular “character,” of 
this or that particular organism. From this assumption we are 
apt to go on to the drawing of new conclusions or the framing of 
new theories as to the ancestral history, the classificatory position, 
the natural affinities of the several organisms: in fact, to apply 
our embryological knowledge mainly, and at times exclusively, to 
the study of phylogeny. When we find, as we are not long of 
finding, that our phylogenetic hypotheses, as drawn from em- 
bryology, become complex and unwieldy, we are nevertheless 
reluctant to admit that the whole method, with its fundamental 
postulates, is at fault. And yet nothing short of this would 
seem to be the case, in regard to the earlier phases at least of 
embryonic development. All the evidence at hand goes, as it 
seems to me, to shew that embryological data, prior to and even 
long after the epoch of segmentation, are essentially a subject for 
physiological and physical investigation and have but the very 
slightest link with the problems of systematic or zoological 
classification. Comparative embryology has its own facts to 
classify, and its own methods and principles of classification. 
Thus we may classify eggs according to the presence or absence, 
the paucity or abundance, of their associated food-yolk, the 
chromosomes according to their form and their number, the 
segmentation according to its various “types,” radial, bilateral, 
spiral, and so forth. But we have little night to expect, and in 
point of fact we shall very seldom and (as it were) only accidentally 
find, that these embryological categories coincide with the lines 
of ‘‘natural’’ or “phylogenetic” classification which have been 
arrived at by the systematic zoologist. 


The cell, which Goodsir spoke of as a “centre of force,” is in 


Iv] STRUCTURE OF THE CELL 197 


‘ 


reality a “sphere of action” of certain more or less localised 
forces; and of these, surface-tension is the particular force which 
is especially responsible for giving to the cell its outline and its 
morphological individuality. The partially segmented differs from 
the totally segmented egg, the unicellular Infusorian from the 
minute multicellular Turbellarian, in the intensity and the range of 
those surface-tensions which in the one case succeed and in the 
other fail to form a visible separation between the “cells.” Adam 
Sedgwick used to call attention to the fact that very often, even 
in eggs that appear to be totally segmented, it is yet impossible 
to discover an actual separation or cleavage, through and through 
between the cells which on the surface of the egg are so clearly 
delimited; so far and no farther have the physical forces effect- 
uated a visible “cleavage.” The vacuolation of the protoplasm in 
Actinophrys or Actinosphaerium is due to localised surface-tensions, 
quite irrespective of the multinuclear nature of the latter 
organism. In short, the boundary walls due to surface-tension 
may be present or may be absent with or without the delimi- 
nation of the other specific fields of force which are usually 
correlated with these boundaries and with the independent 
individuality of the cells. What we may safely admit, however, 
is that one effect of these circumscribed fields of force is usually 
such a separation or segregation of the protoplasmic constituents, 
the more fluid from the less fluid and so forth, as to give a field 
where surface-tension may do its work and bring a visible boundary 
into being. When the formation of a “surface” is once effected, 
its physical condition, or phase, will be bound to differ notably 
from that of the interior of the cell, and under appropriate chemical 
conditions the formation of an actual cell-wall, cellulose or other, 
is easily intelligible. To this subject we shall return again, in 
another chapter. 

From the moment that we enter on a dynamical conception 
of the cell, we perceive that the old debates were in vain as to 
what visible portions of the cell were active or passive, living or 
non-living. For the manifestations of force can only be due to 
the interaction of the various parts, to the transference of energy 
from one to another. Certain properties may be manifested, 
certain functions may be carried on, by the protoplasm apart 


198 ON THE INTERNAL FORM AND [CH. 


from the nucleus; but the interaction of the two is necessary, 
that other and more important properties or functions may be 
manifested. We know, for instance, that portions of an Infusorian 
are incapable of regenerating lost parts in the absence of a nucleus, 
while nucleated pieces soon regain the specific form of the organism : 
and we are told that reproduction by fission cannot be initiated, 
though apparently all its later steps can be carried on, indepen- 
dently of nuclear action. Nor, as Verworn pointed out, can the 
nucleus possibly be regarded as the “sole vehicle of inheritance,” 
since only in the conjunction of cell and nucleus do we find the 
essentials of cell-life. “Kern und Protoplasma sind nur vereint 
lebensfihig,” as Nussbaum said. Indeed we may, with EH. B. 
Wilson, go further, and say that “the terms ‘nucleus’ and ‘cell- 
body’ should probably be regarded as only topographical expres- 
sions denoting two differentiated areas in a common structural 
basis.” 

Endless discussion has taken place regarding the centrosome, 
some holding that it is a specific and essential structure, a per- 
manent corpuscle derived from a similar pre-existing corpuscle, a 
“‘fertilising element” in the spermatozoon, a special “organ of 
cell-division,”” a material “dynamic centre” of the cell (as Van 
Beneden and Boveri call it); while on the other hand, it is pointed 
out that many cells live and multiply without any visible centro- 
somes, that a centrosome may disappear and be created anew, 
and even that under artificial conditions abnormal chemical 
stimuli may lead to the formation of new centrosomes... We may 
safely take it that the centrosome, or the “attraction sphere,’ 
is essentially a “centre of force,” and that this dynamic centre 
may or may not be constituted by (but will be very apt to produce) 
a concrete and visible concentration of matter. 

It is far from correct to say, as is often done, that the cell-wall, 
or cell-membrane, belongs “to the passive products of protoplasm 
rather than to the living cell itself’; or to say that in the animal 
cell, the cell-wall, because it is “‘shghtly developed,” is relatively 
unimportant compared with the important role which it assumes 
in plants. On the contrary, it is quite certain that, whether 
visibly differentiated into a semi-permeable membrane, or merely 
constituted by a liquid film, the surface of the cell is the seat of 


Iv] STRUCTURE OF THE CELL £99 


important forces, capillary and electrical, which play an essential 
part in the dynamics of the cell. Even in the thickened, largely 
solidified cellulose wall of the plant-cell, apart from the mechanical 
resistances which it affords, the osmotic forces developed in con- 
nection with it are of essential importance. 

But if the cell acts, after this fashion, as a whole, each part 
interacting of necessity with the rest, the same is certainly true 
of the entire multicellular organism: as Schwann said of old, in 
very precise and adequate words, “the whole organism subsists 
only by means of the reciprocal action of the single elementary 
parts*.”’ 

As Wilson says again, “the physiological autonomy of the 
individual cell falls into the background...and the apparently 
composite character which the multicellular organism may exhibit 
is owing to a secondary distribution of its energies among local 
centres of actiony.” 

It is here that the homology breaks down which is so often 
drawn, and overdrawn, between the unicellular organism and the 
individual cell of the metazoont. 

Whitman, Adam Sedgwick§, and others have lost no 
opportunity of warning us against a too literal acceptation 
of the cell-theory, against the view that the multicellular 
organism is a colony (or, as Haeckel called it (in the case 
of the plant), a “republic”) of independent units of life||. 
As Goethe said long ago, “Das lebendige ist zwar in Elemente 

* Theory of Cells, p. 191. 

+ The Cell in Development, etc. p. 59; cf. pp. 388, 413. , 

{ E.g. Briicke. Hlementarorganismen, p. 387: ‘Wir miissen in der Zelle einen 
kleinen Thierleib sehen, und diirfen die Analogien, welche zwischen ihr und den 
kleinsten Thierformen existiren, niemals aus den Augen lassen.” 

§ Whitman, C. O., The Inadequacy of the Cell-theory, Journ. of Morphol. 
vill, pp. 639-658, 1893; Sedgwick, A., On the Inadequacy of the Cellular Theory 
of Development, Q.J.M.S. xxxvu, pp. 87-101, 1895, xxxvm1, pp. 331-337, 1896. 
Cf. Bourne, G. C., A Criticism of the Cell-theory; being an answer to Mr Sedewick’s 
article, ete., ibid. xxxvin, pp. 137-174, 1896. 

|| Cf. Hertwig, O., Die Zelle und die Gewebe, 1893, p. 1; “Die Zellen, in welche 
der Anatom die pflanzlichen und thierischen Organismen zerlegt, sind die Trager 
der Lebensfunktionen; sie sind, wie Virchow sich ausgedriickt hat, die ‘Lebensein- 
heiten.’ Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebens- 
process eines zusammengesetzten Organismus nichts Anderes zu sein als das héchst 


verwickelte Resultat der einzelnen Lebensprocesse seiner zahlreichen, verschieden 
functionirenden Zellen.”’ 


200 INTERNAL FORM AND STRUCTURE OF CELL [cu. tv 


zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen 
und beleben;” the dictum of the Cellularpathologie being just 
the opposite, “Jedes Thier erscheint als eine Summe vitaler 
Kinheiten, von denen jede den vollen Charakter des Lebens an 
sich tragt.” 

Hofmeister and Sachs have taught us that in the plant the 
srowth of the mass, the growth of the organ, is the primary fact, 
that “cell formation is a phenomenon very general in organic. 
life, but still only of secondary significance.” ‘Comparative 
embryology” says Whitman, “reminds us at every turn that the 
organism dominates cell-formation, using for the same purpose 
one, several, or many cells, massing its material and directing its 
movements and shaping its organs, as if cells did not exist*.”’ 
So Rauber declared that, in the whole world of organisms, “das 
Ganze liefert die Theile, nicht die Theile das Ganze: letzteres 
setzt die Theile zusammen, nicht diese jenesy.’”’ And on the 
botanical side De Bary has summed up the matter in an aphorism, 
‘‘Die Pflanze bildet Zellen, nicht die Zelle bildet Pflanzen.” 

Discussed almost wholly from the concrete, or morphological 
point of view, the question has for the most part been made to turn 
on whether actual protoplasmic continuity can be demonstrated 
between one cell and another, whether the organism be an actual 
reticulum, or syncytium. But from the dynamical point of view 
the question is much simpler. We then deal not with material 
continuity, not with little bridges of connecting protoplasm, but 
with a continuity of forces, a comprehensive field of force, which 
runs through and through the entire organism and is by no means 
restricted in its passage to a protoplasmic continuum. And such 
a continuous field of force, somehow shaping the whole organism, 
independently of the number, magnitude and form of the individual 
cells, which enter, like a froth, into its fabric, seems to me certainly 
and obviously to exist. As Whitman says, “the fact that physio- 
logical unity is not broken by cell-boundaries is confirmed in so 
many ways that it must be accepted as one of the fundamental 
truths of biolcgyt.” 


* Journ. of Morph. vin, p. 653, 1893. 

y+ Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. vim, pp. 272, 
313, 333, 1883. 

t Journ. of Morph. u, p. 49, 1889. 


CHAPTER V 
THE FORMS OF CELLS 


Protoplasm, as.we have already said, is a fluid or rather a 
semifluid substance, and we need not pause here to attempt to 
describe the particular properties of the semifluid, colloid, or 
jelly-like substances to which it is allied; we should find it no 
easy matter. Nor need we appeal to precise theoretical definitions 
of fluidity, lest we come into a debateable land. It is in the most 
general sense that protoplasm is “fluid.” As Graham said (of 
colloid matter in general), “its softness partakes of fluidity, and 
enables the colloid to become a vehicle for liquid diffusion, like 
water itself*.”” When we can deal with protoplasm in sufficient 
quantity we see it flow; particles move freely through it, air- 
bubbles and liquid droplets shew round or spherical within it; 
and we shall have much to say about other phenomena manifested 
by its own surface, which are those especially characteristic of 
hquids. It may encompass and contain solid bodies, and it may 
“secrete”? within or around itself solid substances; and very 
often in the complex living organism these solid substances 
formed by the living protoplasm, like shell or nail or horn or 
feather, may remain when the protoplasm which formed them 
is dead and gone; but the protoplasm itself is fluid or semifluid, 
and accordingly permits of free (though not necessarily rapid) 
diffusion and easy convection of particles within itself. This simple 
fact is of elementary importance in connection with form, and 
with what appear at first sight to be common characteristics or 
peculiarities of the forms of living things. 

The older naturalists, in discussing the differences between 
inorganic and organic bodies, laid stress upon the fact or state- 
ment that the former grow by “agglutination,” and the latter by 


* Phil. Trans. CLI, p. 183, 1861; Researches, ed. Angus Smith, 1877, p. 553. 


202 THE FORMS OF CELLS [CH. 


what they termed “intussusception.”” The contrast is true, 
rather, of solid as compared with jelly-like bodies of all kinds, 
living or dead, the great majority of which as it so happens, but 
by no means all, are of organic origin. 

A crystal “grows” by deposition of new molecules, one by 
one and layer by layer, superimposed or aggregated upon the 
solid substratum already formed. Each particle would seem to 
be influenced, practically speaking, only by the particles in its 
immediate neighbourhood, and to be in a state of freedom and 
independence from the influence, either direct or indirect, of its 
remoter neighbours. As Lord Kelvin and others have explained 
the formation and the resulting forms of crystals, so we believe 
that each added particle takes up its position in relation to its 
immediate neighbours already arranged, generally in the holes and 
corners that their arrangement leaves, and in closest contact with 
the greatest number*. And hence we may repeat or imitate this 
process of arrangement, with great or apparently even with 
precise accuracy (in the case of the simpler crystalline systems), 
by piling up spherical pills or grains of shot. In so doing, we must 
have regard to the fact that each particle must drop into the 
place where it can go most easily, or where no easier place offers. 
In more technical language, each particle is free to take up, and 
does take up, its position of least potential energy relative to those 
already deposited; in other words, for each particle motion is 
induced until the energy of the system is so distributed that no 
tendency or resultant force remains to move it more. The 
application of this principle has been shewn to lead to the produc- 
tion of planes} (in all cases where by the limitation of material, 
surfaces must occur); and where we have planes, straight edges 
and solid angles must obviously also occur; and, if equilibrium is 


* Cf. Kelvin, On the Molecular Tactics of a Crystal, The Boyle Lecture, Oxford, 
1893, Baltimore Lectures, 1904, pp. 612-642. Here Kelvin was mainly following 
Bravais’s (4nd Frankenheim’s) theory of “space-lattices,”’ but he had been largely 
anticipated by the crystallographers. For an account of the development of the 
subject in modern crystallography, by Sohncke, von Fedorow, Schénfliess, Barlow 
and others, see Tutton’s Crystallography, chap. ix, pp. 118-134, 1911. 

+ In a homogeneous crystalline arrangement, symmetry compels a locus of one 
property to be a plane or set of planes; the locus in this case being that of least 
surface potential energy. 


vy] OF CRYSTALLINE AND COLLOID BODIES — 203 


to follow, must occur symmetrically. Our piling up of shot, or 
manufacture of mimic crystals, gives us visible demonstration 
that the result is actually to obtain, as in the natural crystal, 
plane surfaces and sharp angles, symmetrically disposed. 

But the living cell grows in a totally different way, very much 
as a piece of glue swells up in water, by “imbibition,” or by inter- 
penetration into and throughout its entire substance. The semi- 
fluid colloid mass takes up water, partly to combine chemically 
with its individual molecules*, partly by physical diffusion into 
the interstices between these molecules, and partly, as it would 
seem, in other ways; so that the entire phenomenon is a very 
complex and even an obscure one. But, so far as we are con- 
cerned, the net result is a very simple one. For the equilibrium or 
tendency to equilibrium of fluid pressure in all parts of its interior 
while the process of imbibition is going on, the constant rearrange- 
ment of its fluid mass, the contrast in short with the crystalline 
method of growth where each particle comes to rest to move 
(relatively to the whole) no more, lead the mass of jelly to swell 
up, very much as a bladder into which we blow air, and so, by 
a graded and harmonious distribution of forces, to assume every- 
where a rounded and more or less bubble-like external formf. 
So, when the same school of older naturalists called attention to 
a new distinction or contrast of form between the organic and 
inorganic objects, in that the contours of the former tended to 
roundness and curvature, and those of the latter to be bounded 
by straight lines, planes and sharp angles, we see that this contrast 
was not a new and different one, but only another aspect of 
their former statement, and an immediate consequence of the 
difference between the processes of agglutination and intussus- 
ception. 

This common and general contrast between the form of the 
erystal on the one hand, and of the colloid or of the organism on 
the other, must by no means be pressed too far. For Lehmann, 


* This is what Graham called the water of gelatination, on the analogy of water 
of crystallisation; Chem. and Phys. Researches, p. 597. 

+ Here, in a non-crystalline or random arrangement of particles, symmetry 
ensures that the potential energy shall be the same per unit area of all surfaces; 
and it follows from geometrical considerations that the total surface energy will 
be least if the surface be spherical. 


204 THE FORMS OF CELLS [CH. 


in his great work on so-called Fluid Crystals*, to which we shall 
afterwards return, has shewn how, under certain circumstances, 
surface-tension phenomena may coexist with crystallisation, and 
produce a form of minimal potential which is a resultant of both: 
the fact being that the bonds maintaining the crystalline arrange- 
ment are now so much looser than in the solid condition that the 
tendency to least total surface-area is capable of being satisfied. 
Thus the phenomenon of “liquid crystallisation” does not destroy 
the distinction between crystalline and colloidal forms, but gives 
added unity and continuity to the whole series of phenomena f. 
Lehmann has also demonstrated phenomena within the crystal, 
known for instance as transcrystallisation, which shew us that we 
must not speak unguardedly of the growth of crystals as limited 
to deposition upon a surface, and Biitschli has already pointed out 
the possible great importance to the biologist of the various 
phenomena which Lehmann has describedt. 
So far then, as growth goes on, unaffected by pressure or other 
external force, the fluidity of protoplasm, its mobility internal 
and external, and the manner in which particles move with 
comparative freedom from place to place within, all manifestly 
tend to the production of swelling, rounded surfaces, and to their 
great predominance over plane surfaces in the contour of the 
organism. These rounded contours will tend to be preserved, for 
a while, in the case of naked protoplasm by its viscosity, and in 
. the presence of a cell-wall by its very lack of fluidity. In a general 
way, the presence of curved boundary surfaces will be especially 
obvious in the unicellular organisms, and still more generally in 
the external forms of all organisms; and wherever mutual pressure 
between adjacent cells, or other adjacent parts, has not come into 
play to flatten the rounded surfaces into planes. 
But the rounded contours that are assumed and exhibited by 

* Lehmann, O., Fliissige Krystalle, sowie Plasticitat von Krystallen im allge- 
meinen, etc., 264 pp. 39 pll., Leipsig, 1904. For a semi-popular, illustrated account, 
see Tutton’s Crystals (Int. Sci. Series), 1911. 

+ As Graham said of an allied phenomenon (the so-called blood-crystals of 
Funke), it “illustrates the maxim that in nature there are no abrupt transitions, 
and that distinctions of class are never absolute.” 

t Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik der Organ- 


ismen, Arch. f. Entw. Mech. xx11, p. 207, 1906 (with copious bibliography) ; Lehmann, 
Scheinbar lebende Kristalle und Myelinformen, ibid. xxvi, p. 483, 1908. 


v] OF SURFACE TENSION — 205 


a piece of hard glue, when we throw it into water and see it expand 
as it sucks the water up, are not nearly so regular or so beautiful 
as are those which appear when we blow a bubble, or form a 
drop, or pour water into a more or less elastic bag. For these 
curving contours depend upon the properties of the bag itself, 
of the film or membrane that contains the mobile gas, or that 
contains or bounds the mobile liquid mass. And hereby, in the 
case of the fluid or semifluid mass, we are introduced to the 
subject of surface tension: of which indeed we have spoken in 
the preceding chapter, but which we must now examine with 
greater care. 


Among the forces which determine the forms of cells, whether 
they be solitary or arranged in contact with one another, this 
force of surface-tension is certainly of great, and is probably of 
paramount importance. But while we shall try to separate out 
the phenomena which are directly due to it, we must not forget 
that, in each particular case, the actual conformation which we 
study may be, and usually is, the more or less complex resultant 
of surface tension acting together with gravity, mechanical 
pressure, osmosis, or other physical forces. 

Surface tension is that force by which we explain the form of 
a drop or of a bubble, of the surfaces external and internal of 
a “froth” or collocation of bubbles, and of many other things of 
like nature and in like circumstances*. It is a property of liquids 
(in the sense at least with which our subject is concerned), and it 
is manifested at or very near the surface, where the liquid comes 
into contact with another liquid, a solid or a gas. We note here 
that the term swrface is to be interpreted in a wide sense; for 
wherever we have solid particles imbedded in a fluid, wherever 
we have a non-homogeneous fluid or semi-fluid such as a particle 
* The idea of a “surface-tension”’ in liquids was first enunciated by Segner, 
De figuris superficierum fluidarum, in Comment. Soc. Roy. Gottingen, 1751, p. 301. 
Hooke, in the Micrographia (1665, Obs. vim, etc.), had called attention to the 
globular or spherical form of the little morsels of steel struck off by a flint, and had 
shewn how to make a powder of such spherical grains, by heating fine filings to 
melting point. “This Phaenomenon” he said “proceeds from a propriety which 
belongs to all kinds of fluid Bodies more or less, and is caused by the Incongruity 


of the Ambient and included Fluid, which so.acts and modulates each other, that 
they acquire, as neer as is possible, a spherical or globular form....”’ 


206 THE FORMS OF CELLS [CH. 


of protoplasm, wherever we have the presence of “impurities,” as - 
in a mass of molten metal, there we have always to bear in mind 
the existence of “surfaces” and of surface tensions, not only 
on the exterior of the mass but also throughout its interstices, 
wherever like meets unlike. 

Surface tension is due to molecular force, to force that is to 
say arising from the action of one molecule upon another, and it 
is accordingly exerted throughout a small thickness of material, 
comparable to the range of the molecular forces. We imagine 
that within the interior of the liquid mass such molecular inter- 
actions negative one another: but that at and near the free 
surface, within a layer or film approximately equal to the range 
of the molecular force, there must be a lack of such equilibrium 
and consequently a manifestation of force. 

The action of the molecular forces has been variously explained. 
But one simple explanation (or mode of statement) is that the 
molecules of the surface layer (whose thickness is definite and 
constant) are being constantly attracted into the interior by those 
which are more deeply situated, and that consequently, as 
molecules keep quitting the surface for the interior, the bulk of 
the latter increases while the surface diminishes; and the process 
continues till the surface itself has become a minimum, the swrface- 
shrinkage exhibiting itself as a surface-tension. This is a sufficient 
description of the phenomenon in cases where a portion of liquid 
is subject to no other than its own molecular forces, and (since the 
sphere has, of all solids, the smallest surface for a given volume) 
it accounts for the spherical form of the raindrop, of the grain 
of shot, or of the living cellin many simple organisms. It accounts 
also, as we shall presently see, for a great number of much more 
complicated forms, manifested under less simple conditions. 

Let us here briefly note that surface tension is, in itself, a 
comparatively small force, and easily measurable: for instance 
that of water is equivalent to but a few grains per linear inch, 
or a few grammes per metre. But this small tension, when it 
exists in a curved surface of very great curvature, gives rise to a 
very great pressure directed towards the centre of curvature. We 
can easily calculate this pressure, and so satisfy ourselves that, 
when the radius of curvature is of molecular dimensions, the 


' 


v] . OF SURFACE ENERGY 207 


_ pressure is of the magnitude of thousands of atmospheres,—a con- 
clusion which is supported by other physical considerations. 

The contraction of a liquid surface and other phenomena of 
surface tension involve the doing of work, and the power to do 
work is what we call energy. It is obvious, in such a simple case 
as we have just considered, that the whole energy of the system 
is diffused throughout its molecules; but of this whole stock of 
energy it is only that part which comes into play at or very near 
to the surface which normally manifests itself in work, and hence 
we may speak (though the term is open to some objections) of 
a specific surface energy. The consideration of surface energy, 
and of the manner in which its amount is increased and multiplied 
by the multiplication of surfaces due to the subdivision of the 
organism into cells, is of the highest importance to the physiologist ; 
and even the morphologist cannot wholly pass it by, if he desires 
to study the form of the cell in its relation to the phenomena of 
surface tension or “‘capillarity.” The case has been set forth with 
the utmost possible lucidity by Tait and by Clerk Maxwell, on 
whose teaching the following paragraphs are based: they having 
based their teaching upon that of Gauss,—who rested on Laplace. 

Let # be the whole potential energy of a mass M of liquid; 
let e, be the energy per unit mass of the interior liquid (we may 
call it the internal energy); and let e be the energy per unit mass 
for a layer of the skin, of surface S, of thickness t, and density 
p (e being what we call the surface energy). It is obvious that the 
total energy consists of the internal plus the surface energy, and 
that the former is distributed through the whole mass, minus its 
surface layers. That is to say, in mathematical language, 


E=(M—S. Xtp)eg +S. Xtpe. 
But this is equivalent to writing: 
= Me,+ 8S. dip (e — e&); 


and this is as much as to say that the total energy of the system 
may be taken to consist of two portions, one uniform throughout 
the whole mass, and another, which is proportional on the one hand 
to the amount of surface, and on the other hand is proportional 
to the difference between e and ey, that is to say to the difference 
between the unit values of the internal and the surface energy. 


208 THE FORMS OF CELLS [CH. 


It was Gauss who first shewed after this fashion how, from 
the mutual attractions between all the particles, we are led to an 
expression which is what we now call the potential energy of the 
system; and we know, as a fundamental theorem of dynamics, 
that the potential energy of the system tends to a minimum, and 
in that minimum finds, as a matter of course, its stable equilibrium. 


We see in our last equation that the term Me, is irreducible, 
save by a reduction of the mass itself. But the other term may 
be diminished (1) by a reduction in the area of surface, S, or 
(2) by a tendency towards equality of e and é9, that is to say by 
a diminution of the specific surface energy, e. 

These then are the two methods by which the energy of the 
system will manifest itself in work. The one, which is much the 
more important for our purposes, leads always to a diminution of 
surface, to the so-called “principle of minimal areas”; the other, 
which leads to the lowering (under certain circumstances) of 
surface tension, is the basis of the theory of Adsorption, to which 
we shall have some occasion to refer as the modus operandi in the 
development of a cell-wall, and in a variety of other histological 
phenomena. In the technical phraseology of the day, the 
‘capacity factor” is involved in the one case, and the “intensity 
factor”? in the other. 

Inasmuch as we are concerned with the form of the cell it is 
the former which becomes our main postulate: telling us that 
the energy equations of the surface of a cell, or of the free surfaces 
of cells partly in contact, or of the partition-surfaces of cells in 
contact with one another or with an adjacent solid, all indicate 
a minimum of potential energy in the system, by which the system 
is brought, ipso facto, into equilibrium. And we shall not fail to 
observe, with something more than mere historical interest and 
curiosity, how deeply and intrinsically there enter into this whole 
class of problems the “principle of least action” of Maupertuis, 
the “lineae curvae maximi minimive proprietate gaudentes” of 
Euler, by which principles these old natural philosophers explained 
correctly a multitude of phenomena, and drew the lines whereon 
the foundations of great part of modern physics are well and 
truly laid. 


v] THE MEANING OF SYMMETRY 209 


In all cases where the principle of maxima and minima comes 
into play, as it conspicuously does in the systems of liquid films 
which are governed by the laws of surface-tension, the figures and 
conformations produced are characterised by obvious and remark- 
able symmetry. Such symmetry is in a high degree characteristic 
of organic forms, and is rarely absent in living things,—save in such 
cases as amoeba, where the equilibrium on which symmetry depends 
is likewise lacking. And if we ask what physical equilibrium has 
to do with formal symmetry and regularity, the reason is not far 
to seek; nor can it be put_better than in the following words of 
Mach’s*. “In every symmetrical system every deformation that 
tends to destroy the symmetry is complemented by an equal and 
opposite deformation that tends to restore it. In each deformation 
positive and negative work is done. One condition, therefore, 

‘though not an absolutely sufficient one, that a maximum or 
minimum of work corresponds to the form of equilibrium, is thus 
supplied by symmetry. Regularity is successive symmetry. 
There is no reason, therefore, to be astonished that the forms of 
equilibrium are often symmetrical and regular.” 


As we proceed in our enquiry, and especially when we approach 
the subject of t¢sswes, or agglomerations of cells, we shall have 
from time to time to call in the help of elementary mathematics. 
But already, with very little mathematical help, we find ourselves 
in a position to deal with some simple examples of organic forms. 

When we melt a stick of sealing-wax in the flame, surface 
tension (which was ineffectively present in the solid but finds play 
in the now fluid mass), rounds off its sharp edges into curves, so 
striving towards a surface of minimal area; and in like manner, 
by melting the tip of a thin rod of glass, Leeuwenhoek made the 
little spherical beads which served him for a microscopet. When 
any drop of protoplasm, either over all its surface or at some free 
end, as at the extremity of the pseudopodium of an amoeba, is 


* Science of Mechanics, 1902, p. 395; see also Mach’s article Ueber die physika- 
lische Bedeutung der Gesetze der Symmetrie, Lotos, xx1, pp. 139-147, 1871. 

+ Similarly, Sir David Brewster and others made powerful lenses by simply 
dropping small drops of Canada balsam, castor oil, or other strongly refractive 
liquids, on to a glass plate: On New Philosophical Instruments (Description of a 
new Fluid Microscope), Edinburgh, 1813, p. 413. 


ath (ee 14 


210 THE FORMS OF CELLS [on. 


4 


seen likewise to “round itself off,” that is not an effect of “vital 
contractility,’ but (as Hofmeister shewed so long ago as 1867) 
~ a simple consequence of surface tension; and almost immediately 
afterwards Engelmann* argued on the same lines, that the forces 
which cause the contraction of protoplasm in general may “be 
just the same as those which tend to make every non-spherical 
drop of fluid become spherical!”” We are not concerned here with 
the many theories and speculations which would connect the 
phenomena of surface tension with contractility, muscular move- 
ment or other special physiological functions, but we find ample 
room to trace the operation of the same cause in producing, under 
conditions of rest and equilibrium, certain definite and inevitable 
forms of surface. 

It is however of great importance to observe that the living 
cell is one of those cases where the phenomena of surface tension 
are by no means limited to the owter surface; for within the 
heterogeneous substance of the cell, between the protoplasm and 
its nuclear and other contents, and in the alveolar network of the 
cytoplasm itself (so far as that “alveolar structure” is actually 
present in life), we have a multitude of interior surfaces; and, 
especially among plants, we may have a large, inner surface of 
“interfacial”? contact, where the protoplasm contains cavities 
or “vacuoles” filled with a different and more fluid material, the 
“cell-sap.” Here we have a great field for the development of 
surface tension phenomena: and so long ago as 1865, Nageli and 
Schwendener shewed that the streaming currents of plant cells 
might be very plausibly explained by this phenomenon. Even 
ten years earlier, Weber had remarked upon the resemblance 
between these protoplasmic streamings and the streamings to be 
observed in certain inanimate drops, for which no cause but 
surface tension could be assigned f. 

The case of amoeba, though it is an elementary case, is at the 
same time a complicated one. While it remains “amoeboid,” it 
is never at rest or in equilibrium; it is always moving, from one 
to another of its protean changes of configuration; its surface 
tension is constantly varying from point to point. Where the 


* Beitrige z. Physiologie d. Protoplasma, Pfliiger’s Archiv, 11, p. 307, 1869. 
t Poggend. Annalen, xciv, pp. 447-459, 1855. Cf. Strethill Wright, Phil. 
Mag. Feb. 1860. 


v] THE FORM OF AMOEBA 211 


surface tension is greater, that portion of the surface will contract 
into spherical or spheroidal forms; where it is less the surface 
will correspondingly extend. While generally speaking the surface 
energy has a minimal value, it is not necessarily constant. It may 
be diminished by a rise of temperature; it may be altered by 
contact with adjacent substances*, by the transport of constituent 
materials from the interior to the surface, or again by actual 
chemical and fermentative change. Within the cell, the surface 
energies developed about its heterogeneous contents will constantly 
vary as these contents are affected by chemical metabolism. As 
the colloid materials are broken down and as the particles in 
suspension are diminished in size the “free surface energy” 
will be increased, but the osmotic energy will be diminished f. 
Thus arise the various fluctuations of surface tension and the 
various phenomena of amoeboid form and motion, which Biitschli 
and others have reproduced or imitated by means of the fine 
emulsions which constitute their “artificial amoebae.” A multi- 
tude of experiments shew how extraordinarily delicate is the 
adjustment of the surface tension forces, and how sensitive they 
are to the least change of temperature or chemical state. Thus, 
on a plate which we have warmed at one side, a drop of alcohol 
runs towards the warm area, a drop of oil away from it; and a 
drop of water on the glass plate exhibits lively movements when 


* Hayceraft and Carlier pointed out (Proc. R.S.H. xv, pp. 220-224, 1888) that 
the amoeboid movements of a white blood-corpuscle are only manifested when the 
corpuscle is in contact with some solid substance: while floating freely in the 
plasma or serum of the blood, these corpuscles are spherical, that is to say they 
are at rest and in equilibrium. The same fact has recently been recorded anew 
by Ledingham (On Phagocytosis from an adsorptive point of view, Journ. of Hygiene, 
xu, p. 324, 1912). On the emission of pseudopodia as brought about by changes 
in surface tension, see also (int. al.) Jensen, Ueber den Geotropismus niederer 
Organismen, Pfliiger’s Archiv, Lim, 1893. Jensen remarks that in Orbitolites, the 
pseudopodia issuing through the pores of the shell first float freely, then as they 
grow longer bend over till they touch the ground, whereupon they begin to display 
amoeboid and streaming motions. Verworn indicates (Allg. Physiol. 1895, p. 429), 
and Davenport says (Hxperim. Morphology, 11, p. 376) that “this persistent clinging 
to the substratum is a ‘thigmotropic’ reaction, and one which belongs clearly to 
the category of ‘response.’”’ (Cf. Piitter, Thigmotaxis bei Protisten, A. f. Physiol. 
1900, Suppl. p. 247.) But it is not clear to my mind that to account for this 
simple phenomenon we need invoke other factors than gravity and surface-action. 

+ Cf. Pauli, Allgemeine physikalische Chemie d. Zellen u. Gewebe, in Asher-Spiro’s 
Ergebnisse der Physiologie, 1912; Przibram, Vitalitdt, 1913, p. 6. 


14—2 


212 THE FORMS OF CELLS [CH. 


we bring into its neighbourhood a heated wire, or a glass rod 
dipped in ether. When we find that a plasmodium of Aethalium, 
for instance, creeps towards a damp spot, or towards a warm spot, 
or towards substances that happen to be nutritious, and again 
creeps away from solutions of sugar or of salt, we seem to be 
dealing with phenomena every one of which can be paralleled by 
ordinary phenomena of surface tension*. Even the soap-bubble 
itself is imperfectly in equilibrium, for the reason that its film, 
like the protoplasm of amoeba or Aethalium, is an excessively 
heterogeneous substance. Its surface tensions vary from point 
to poimt, and chemical changes and changes of temperature 
increase and magnify the variation. The whole surface of the 
bubble is in constant movement as the concentrated portions of 
the soapy fluid make their way outwards from the deeper layers; 
it thins and it thickens, its colours change, currents are set up in 
it, and little bubbles glide over it; it continues in this state of 
constant movement, as its parts strive one with another in all 
their interactions towards equilibrium fF. 

In the case of the naked protoplasmic cell, as the amoeboid 
phase is emphatically a phase of freedom and activity, of chemical 
and physiological change, so, on the other hand, is the spherical 
form indicative of a phase of rest or comparative inactivity. In 
the one phase we see unequal surface tensions manifested in the 
creeping movements of the amoeboid body, in the rounding off 
of the ends of the pseudopodia, in the flowing out of its substance 
over a particle of “food,” and in the current-motions in the interior 
of its mass; till finally, in the other phase, when internal homo- 
geneity and equilibrium have been attained and the potential 


* The surface-tension theory of protoplasmic movement has been denied by 
many. Cf. (e.g.), Jennings, H. §., Contributions to the Study of the Behaviour 
of the Lower Organisms, Carnegie Inst. 1904, pp. 130-230; Dellinger, O. P., 
Locomotion of Amoebae, ete. Journ. Exp. Zool. m1, pp. 337-357, 1906; also various 
papers by Max Heidenhain, in Anatom. Hefte (Merkel und Bonnet), ete. 

+ These various movements of a liquid surface, and other still more striking 
movements such as those of a piece of camphor floating on water, were at one time 
ascribed by certain physicists to a peculiar force, swi generis, the force épipolique 
of Dutrochet: until van der Mensbrugghe shewed that differences of surface tension 
were enough to account for this whole series of phenomena (Sur la tension super- 
ficielle des liquides considérée au. point de vue de certains mouvements observés 
a leur surface, Mém. Cour. Acad. de Belgique, xxxtv, 1869; cf. Plateau, p. 283). 


v] ' THE FORM OF AMOEBA 213 


energy of the system is for the time being at a minimum, the 
cell assumes a rounded or spherical form, passing into a state 
of “rest,” and (for a reason which we shall presently see) 
becoming at the same time “encysted.”’ 

In a budding yeast-cell (Fig. 60), we see a more definite and 
restricted change of surface tension. When a “bud” appears, 
whether with or without actual growth by osmosis 
or otherwise of the mass, it does so because at a 
certain part of the cell-surface the surface tension 
has more or less suddenly diminished, and the 
area of that portion expands accordingly; but in 
turn the surface tension of the expanded area will \:° 3) 
make itself felt, and the bud will be rounded off Fig. 60. 
into a more or less spherical form. 

The yeast-cell with its bud is a simple example of a principle 
which we shall find to be very important. Our whole treatment 
of cell-form in relation to surface-tension depends on the fact 
(which Errera was the first to point out, or to give clear expression 
to) that the incipient cell-wall retains with but little impairment 
the properties of a liquid film*, and that the growing cell, in spite 
of the membrane by which it has already begun to be surrounded, 
behaves very much like a fluid drop. But even the ordinary 
yeast-cell shows, by its ovoid and non-spherical form, that it has 
acquired its shape under the influence of some force other than 
that uniform and symmetrical surface-tension which would be 
productive of a sphere; and this or any other asymmetrical form, 
once acquired, may be retained by virtue of the solidification and 
consequent rigidity of the membranous wall of the cell. Unless 
such rigidity ensue, it is plain that such a conformation as that of 
the cell with its attached bud could not be long retained, amidst 
the constantly varying conditions, as a figure of even partial 
equilibrium. But as a matter of fact, the cell in this case is not 
in equilibrium at all; it is in process of budding, and is slowly 
altering its shape by rounding off the bud. It is plain that over 
its surface the surface-energies are unequally distributed, owing 
to some heterogeneity of the substance; and to this matter we 
shall afterwards return. In like manner the developing egg 


* Cf. infra, p. 306. 


214 THE FORMS OF CELLS. [cH. 


through all its successive phases of form is never in complete 
equilibrium; but is merely responding to constantly changing 
conditions, by phases of partial, transitory, unstable and con- 
ditional equilibrium. 

It is obvious that there are innumerable solitary plant-cells, 
and unicellular organisms in general, which, like the yeast-cell, do 
not correspond to any of the simple forms that may be generated 
under the influence of simple and homogeneous surface-tension ; 
and in many cases these forms, which we should expect to be 
unstable and transitory, have become fixed and stable by reason 
of the comparatively sudden or rapid solidification of the envelope. 
This is the case, for instance, in many of the more complicated forms 
of diatoms or of desmids, where we are dealing, in a less striking 
but even more curious way than in the budding yeast-cell, not 
with one simple act of formation, but with a complicated result 
uf successive stages of localised growth, interrupted by phases of 
partial consolidation. The original cell has acquired or assumed 
a certain form, and then, under altering conditions and new 
distributions of energy, has thickened here or weakened there, 
and has grown out or tended (as it were) to branch, at particular 
points. We can often, or indeed generally, trace in each particular 
stage of growth or at each particular temporary growing point, 
the laws of surface tension manifesting themselves in what is 
for the time being a fluid surface; nay more, even in the adult 
and completed structure; we have little difficulty in tracing and 
recognising (for instance in the outline of such a desmid as Euas- 
trum) the rounded lobes that have successively grown or flowed 
out from the original rounded and flattened cell. What we see in 
a many chambered foraminifer, such as Globigerina or Rotalia, is 
just the same thing, save that it is carried out in greater complete- 
ness and perfection. The little organism as a whole is not a figure 
of equilibrium or of minimal area; but each new bud or separate 
chamber is such a figure, conditioned by the forces of surface 
tension, and superposed upon the complex aggregate of similar 
bubbles after these latter have become consolidated one by one 
into a rigid system. 


Let us now make some enquiry regarding the various forms 


=p S) Be eae) ge 


v] OF LIQUID FILMS 215 


which, under the influence of surface tension, a surface can possibly 
assume. In doing so, we are obviously limited to conditions 
under which other forces are relatively unimportant, that is to 
say where the “surface energy” is a considerable fraction of 
the whole energy of the system; and this in general will be 
the case when we are dealing with portions of liquid so small 
that their dimensions come within what we have called the 
molecular range, or, more generally, in which the “specific 
surface” is large*: in other words it will be small or minute 
organisms, or the small cellular elements of larger organisms, 
whose forms will be governed by surface-tension; while the 
general forms of the larger organisms will be due to other and 
non-molecular forces. For instance, a large surface of water sets 
itself level because here gravity is predominant; but the surface 
of water in a narrow tube is manifestly curved, for the reason 
that we are here dealing with particles which are mutually within 
the range of each other’s molecular forces. The same is the case 
with the cell-surfaces and cell-partitions which we are presently 
to study, and the effect of gravity will be especially counteracted 
and concealed when, as in the case of protoplasm in a watery 
fluid, the object is immersed in a liquid of nearly its own specific 
gravity. 

We have already learned, as a fundamental law of surface- 
tension phenomena, that a liquid film in equilibrium assumes a 
form which gives it a minimal area under the conditions to which 
it is subject. And these conditions include (1) the form of the 
boundary, if such exist, and (2) the pressure, if any, to which the 
film is subject; which pressure is closely related to the volume, 
of air or of liquid, which the film (if it be a closed one) may have 
to contain. In the simplest of cases, when we take up a soap- 
film on a plane wire ring, the film is exposed to equal atmospheric 
pressure on both sides, and it obviously has its minimal area in 
the form of a plane. So long as our wire ring lies in one plane 
(however irregular in outline), the film stretched across it will 
still be in a plane; but if we bend the ring so that it lies no longer 
in a plane, then our film will become curved into a surface which 
may be extremely complicated, but is still the smallest possible 


* Cf. p. 32. 


216 THE FORMS OF CELLS [CH. 


surface which can be drawn continuously across the uneven 
boundary. 

The question of pressure involves not only external pressures 
acting on the film, but also that. which the film itself is capable 
of exerting. For we have seen that the film is always contracting 
to its smallest limits; and when the film is curved, this obviously 
leads to a pressure directed inwards,—perpendicular, that is to 
say, to the surface of the film. In the case of the soap-bubble, 
the uniform contraction of whose surface has led to its spherical 
form, this pressure is balanced by the pressure of the air within; 
and if an outlet be given for this air, then the bubble contracts 
with perceptible force until it stretches across the mouth of the 
tube, for instance the mouth of the pipe through which we have 
blown the bubble. A precisely similar pressure, directed inwards, 
is exercised by the surface layer of a drop of water or a globule 
of mercury, or by the surface pellicle on a portion or “drop” of 
protoplasm. Only we must always remember that in the soap- 
bubble, or the bubble which a glass-blower blows, there is a twofold 
pressure as compared with that which the surface-film exercises 
on the drop of liquid of which it is a part; for the bubble consists 
(unless it be so thin as to consist of a mere layer of molecules*) 
of a liquid layer, with a free surface within and another without, 
and each of these two surfaces exercises its own independent and 
coequal tension, and corresponding pressure f. 


If we stretch a tape upon a flat table, whatever be the tension 
of the tape it obviously exercises no pressure upon the table 
below. But if we stretch it over a curved surface, a cylinder for 
instance, it does exercise a downward pressure; and the more 
curved the surface the greater is this pressure, that is to say the 
greater is this share of the entire force of tension which is resolved 
in the downward direction. In mathematical language, the 
pressure (p) varies directly as the tension (7), and inversely as 
the radius of curvature (R): that is to say, p = T/R, per unit of 
surface. 


* Or, more strictly speaking, unless its thickness be less than twice the range 
of the molecular forces. 

+ It follows that the tension, depending only on the surface-conditions, is 
independent of the thickness of the film. 


v| OF LIQUID FILMS 217 


If instead of a cylinder, which is curved only in one direction, 
we take a case where there are curvatures in two dimensions (as 
for instance a sphere), then the effects of these must be simply 
added to one another, and the resulting pressure p is equal to 
T/R+T/R' or p=T (1/R + 1/R’)*. 

And if in addition to the pressure p, which is due to surface 
tension, we have to take into account other pressures, p’, p’’, etc., 
which are due to gravity or other forces, then we may say that 
the total pressure, P= p' + p’ + T (1/R+1/R’). While in some 
cases, for instance in speaking of the shape of a bird’s egg, we 
shall have to take account of these extraneous pressures, in the 
present part of our subject we shall for the most part be able to 
neglect them. 

Our equation is an equation of equilibrium. The resistance 
to compression,—the pressure outwards,—of our fluid mass, is a 
constant quantity (P); the pressure inwards, T (1/R + 1/R’), is 
also constant; and if (unlike the case of the mobile amoeba) the 
surface be homogeneous, so that 7 is everywhere equal, it follows 
that throughout the whole surface 1/R + 1/R’ = C (a constant). 

Now equilibrium is attained after the surface contraction has 
done its utmost, that is to say when it has reduced the surface 
to the smallest possible area; and so we arrive, from the physical 
side, at the conclusion that a surface such that 1/R + 1/R’ = C, 
in other words a surface which has the same mean curvature at 
all points, is equivalent to a surface of minimal area: and to the 
same conclusion we may also arrive through purely analytical 
mathematics. It is obvious that the plane and the sphere are two 
examples of such surfaces, for in both cases the radius of curvature 
is everywhere constant, being equal to infinity in the case of the 
plane, and to some definite magnitude in the case of the sphere. 


From the fact that we may extend a soap-film across a ring of 
wire however fantastically the latter may be bent, we realise that 
there is no limit to the number of surfaces of minimal area which 
may be constructed or may be imagined; and while some of these 
are very complicated indeed, some, for instance a spiral helicoid 
screw, are relatively very simple. But if we limit ourselves to 


* This simple but immensely important formula is due to Laplace (Mécanique 
Céleste, Bk x. suppl. Théorie de Vaction capillaire, 1806). 


218 THE FORMS OF CELLS (cH. 


surfaces of revolution (that is to say, to surfaces symmetrical about 
an axis), we find, as Plateau was the first to shew, that those which 
meet the case are very few in number. They are six in all, 
namely the plane, the sphere, the cylinder, the catenoid, the 
unduloid, and a curious surface which Plateau called the nodoid. 

These several surfaces are all closely related, and the passage 
from one to another is generally easy. Their mathematical inter- 
relation is expressed by the fact (first shewn by Delaunay*, in 1841) 
that the plane curves by whose rotation they are generated are 
themselves generated as “roulettes” of the conic sections. 

Let us imagine a straight line upon which a circle, an ellipse 
or other conic section rolls; the focus of the conic section will 
describe a line in some relation to the fixed axis, and this line 
(or roulette), rotating around the axis, will describe in space one or 
other of the six surfaces of revolution with which we are dealing. 

If we imagine an ellipse so to roll over a line, either of its foci’ 
will describe a sinuous or wavy line (Fig. 618) at a distance 


Fig. 61. 


alternately maximal and minimal from the axis; and this wavy 
line, by rotation about the axis, becomes the meridional line of 
the surface which we call the wnduloid. The more unequal the 
two axes are of our ellipse, the more pronounced will be the 
sinuosity of the described roulette. If the two axes be equal, 
then our ellipse becomes a circle, and the path described by its 
rolling centre is a straight line parallel to the axis (A); and 
obviously the solid of revolution generated therefrom will be a 
cylinder. Tf one axis of our ellipse vanish, while the other remain 
of finite length, then the ellipse is reduced to a straight line, and 
its roulette will appear as a succession of semicircles touching one 
another upon the axis (C); the solid of revolution will be a series of 
equal spheres. If as before one axis of the ellipse vanish, but the 
other be infinitely long, then the curve described by the rotation 


* Sur la surface de révolution dont la courbure moyenne est constante, Journ. 
de M. Liouville, v1, p. 309, 1841. 


wal OF MINIMAL SURFACES 219 


of this latter will be a circle of infinite radius, i.e. a straight line 
infinitely distant from the axis; and the surface of rotation is now 
a plane. If we imagine one focus of our ellipse to remain at a 
given distance from the axis, but the other to become infinitely 
remote, that is tantamount to saying that the ellipse becomes 
transformed into a parabola; and by the rolling of this curve 
along the axis there is described a catenary (D), whose solid of 
revolution is the catenoid. 

Lastly, but this is a little more difficult to imagine, we have 
the case of the hyperbola. 

We cannot well imagine the hyperbola rolling upon a fixed 
straight line so that its focus shall describe a continuous curve. 
But let us suppose that the fixed line is, to begin with, asymptotic 
to one branch of the hyperbola, and that the rolling proceed 
until the line is now asymptotic to the other branch, that is to 
say touching it at an infinite distance; there will then be mathe- 
matical continuity if we recommence rolling with this second 
branch, and so in turn with the other, when each has run its 
course. We shall see, on reflection, that the line traced by one 
and the same focus will be an ‘elastic curve” describing a suc- 
cession of kinks or knots (E), and the solid of revolution described 
by this meridional line about the axis is the so-called nodoid. 

The physical transition of one of these surfaces into another 
can be experimentally illustrated by means of soap-bubbles, or 
better still, after the method of Plateau, by means-of a large 
globule of oil, supported when necessary by wire rings, within a 
fluid of specific gravity equal to its own. 

To prepare a mixture of alcohol and water of a density precisely 
equal to that of the oil-globule is a troublesome matter, and a 
method devised by Mr C. R. Darling is a great improvement on 
Plateau’s*. Mr Darling uses the oily liquid orthotoluidene, which 
does not mix with water, has a beautiful and conspicuous red 
colour, and has precisely the same density as water when both 
are kept at a temperature of 24°C. We have therefore only to 
run the liquid into water at this temperature in order to produce 
beautifully spherical drops of any required size: and by adding 


* See Liquid Drops and Globules, 1914, p. 11. Robert Boyle used turpentine 
in much the same way. For other methods see Plateau, op. cit. p. 154. 


220 THE FORMS OF CELLS [CH. 


a little salt to the lower layers of water, the drop may be made 
to float or rest upon the denser liquid. 

We have already seen that the soap-bubble, spherical to begin 
with, is transformed into a plane when we relieve its internal 
pressure and let the film shrink back upon the orifice of the pipe. 
If we blow a small bubble and then catch it up on a second pipe, 
so that it stretches between, we may gradually draw the two pipes 
apart, with the result that the spheroidal surface will be gradually 
flattened in a longitudinal direction, and the bubble will be trans- 
formed into a cylinder. But if we draw the pipes yet farther 
apart, the cylinder will narrow in the middle into a sort of hour- 
glass form, the increasing curvature of its transverse section being 
balanced by a gradually increasing negative curvature in the 
longitudinal section. The cylinder has, in turn, been converted 
into an unduloid. When we hold a portion of a soft glass tube in 
the flame, and “draw it out,” we are in the same identical fashion 


Fig. 62. 


converting a cylinder into an unduloid (Fig. 62 a); when on the 
other hand we stop the end and blow, we again convert the 
cylinder into an unduloid (B), but into one which is now positively, 
while the former was negatively curved. The two figures are 
essentially the same, save that the two halves of the one are 
reversed in the other. 

That spheres, cylinders and unduloids are of the commonest 
occurrence among the forms of small unicellular organism, or of 
individual cells in the simpler aggregates, and that in the processes 
of growth, reproduction and development transitions are frequent 
from one of these forms to another, is obvious to the naturalist, 
and we shall deal presently with a few illustrations of these 
phenomena. 

But before we go further in this enquiry, it will be necessary 
to consider, to some small extent at least, the curvatures of the 
six different surfaces, that is to say, to determine what modification 


v] OF MINIMAL SURFACES 221 


is required, in each case, of the general equation which applies 
to them all. We shall find that with this question is closely 
connected the question of the pressures exercised by, or im- 
pinging on the film, and also the very important question of 
the limitations which, from the nature of the case, exist to 
prevent the extension of certain of the figures beyond certain 
bounds. The whole subject is mathematical, and we shall only 
deal with it in the most elementary way. 

We have seen that, in our general formula, the expression 
1/R + 1/R’ = C,.a constant; and that this is, in all cases, the 
condition of our surface being one of minimal area. In other 
words, it is always true for one and all of the six surfaces which 
we have to consider. But the constant C may have any value, 
positive, negative, or nil. 

_ In the case of the plane, where R and RF’ are both infinite, it 
is obvious that 1/R + 1/R’ = 0. The expression therefore vanishes, 
and our dynamical equation of equilibrium becomes P= p. In 
short, we can only have a plane film, or we shall only find a plane 
surface in our cell, when on either side thereof we have equal 
pressures or no pressure at all. A simple case is the plane partition 
between two equal and similar cells, as in a filament of spirogyra. 

In the case of the sphere, the radii are all equal, R= R’; 
they are also positive, and T (1/R + 1/R’), or 2T/R, is a positive 
quantity, involving a positive pressure P, on the other side of the 
equation. 

In the cylinder, one radius of curvature has the finite and 
positive value R; but the other is infinite. Our formula becomes 
T/R, to which corresponds a positive pressure P, supplied by the 
surface-tension as in the case of the sphere, but evidently of just 
half the magnitude developed in the latter case for a given value 
of the radius R. 

The catenoid has the remarkable property that its curvature in 
one direction is precisely equal and opposite to its curvature in 
the other, this property holding good for all points of the surface. 
That is to say, R = — R’; and the expression becomes 


(1/R + 1/R’) = (1/R — 1/R) =0; 


in other words, the surface, as in the case of the plane, has no 


222 THE FORMS OF CELLS [cH. 


curvature, and exercises no pressure. ‘There are no other surfaces, 
save these two, which share this remarkable property; and it 
follows, as a simple corollary, that we may expect at times to have 
the catenoid and the plane coexisting, as parts of one and the 
same boundary system; just as, in a cylindrical drop or cell, the 
cylinder is capped by portions of spheres, such that the cylindrical 
and spherical portions of the wall exert equal positive pressures. 

In the unduloid, unlike the four surfaces which we have just 
been considering, it is obvious that the curvatures change from 
one point to another. At the middle of one of the swollen 
portions, or “beads,” the two curvatures are both positive; the 
expression (1/R + 1/R’) is therefore positive, and it is also finite. 
The film, accordingly, exercises a positive tension inwards, which 
must be compensated by a finite and positive outward pressure 
P. At the middle of one of the narrow necks, between two 
adjacent beads, there is obviously, in the transverse direction, 
a much-stronger curvature than in the former case, and the curva- 
ture which balances it is now a negative one. But the sum of the 
two must remain positive, as well as constant; and we therefore 
see that the convex or positive curvature must always be greater 
than the concave or negative curvature at the same point. This 
is plainly the case in our figure of the unduloid. 

The nodoid is, like the unduloid, a continuous curve which 
keeps altering its curvature as it alters its distance from the axis; 
but in this case the resultant pressure inwards is negative instead 
of positive. But this curve is a complicated one, and a full 
discussion of it would carry us beyond our scope. 

In one of Plateau’s experiments, a bubble of oil (protected from 
gravity by the specific gravity of the surrounding fluid being 

identical with its own) is balanced between two 
annuli. It may then be brought to assume the form 
of Fig. 63, that is to say the form of a cylinder with 
spherical ends; and there is then everywhere, owing 
to the convexity of the surface film, a pressure 
inwards upon the fluid contents of the bubble. If 
the surrounding liquid be ever so little heavier or 
lighter than that which constitutes the drop, then 


Fig. 63. the conditions of equilibrium will be accordingly 


v] OF FIGURES OF EQUILIBRIUM 223 


modified, and the cylindrical drop will assume the form of an 
unduloid (Fig. 64 a, B), with its dilated portion below or above, 
as the case may be; and our cylinder 

may also, of course, be converted 

into an unduloid either by elongating 

it further, or by abstracting a portion 

of its oil, until at length rupture 

ensues and the cylinder breaks up 

into two new spherical drops. In all 

cases alike, the unduloid, like the A B 
original cylinder, will be capped by ae Oe 
spherical ends, which are the sign, and the consequence, of the 
positive pressure produced by the curved walls of the unduloid. 
But if our initial cylinder, instead of being tall, be a flat or 
dumpy one (with certain definite relations of height to breadth), 
then new phenomena may be exhibited. For now, if a little 
oil be cautiously withdrawn from the mass by help of a small 
syringe, the cylinder may be made to flatten down so that 
its upper and lower surfaces become plane; which is of itself 
an indication that the pressure inwards is now nil. But at 
the very moment when the upper and lower surfaces become 
plane, it will be found that the sides curve inwards, in the 
fashion shewn in Fig. 658. This figure is a catenoid, which, as 


A B 


Fig. 65. 


we have already seen, is, like the plane itself, a surface exercising 
no pressure, and which therefore may coexist with the plane as 
part of one and the same system. We may continue to withdraw 
more oil from our bubble, drop by drop, and now the upper and 
lower surfaces dimple down into concave portions of spheres, as 
the result of the negative internal pressure; and thereupon the 
peripheral catenoid surface alters its form (perhaps, on this small 
scale, imperceptibly), and becomes a portion of a nodoid (Fig. 65 a). 


224 THE FORMS OF CELLS [CH. 


It represents, in fact, that portion of the nodoid, which 
in Fig. 66 lies between such points as 0, Pp. While it is easy to 
draw the outline, or meridional 

M N section, of the nodoid (as in 

Fig. 66), it is obvious that the 

solid of revolution to be derived 

- 0 from it, can never be realised in 
its entirety: for one part of the 
solid figure would cut, or en- 
tangle with, another. All that 
we can ever do, accordingly, is to realise isolated portions of the 
nodoid. 

If, in a sequel to the preceding experiment of Plateau’s, we 
use solid dises instead of annuli, so as to enable us to exert direct 
mechanical pressure upon our globule of oil, we again begin by 
adjusting the pressure of these discs so that the oil assumes the 
form of a cylinder: our discs, that is to say, are adjusted to 
exercise a mechanical pressure equal to what in the former case 
was supplied by the surface-tension of the spherical caps or ends 
of the bubble. If we now increase the pressure slightly, the 
peripheral walls will become convexly curved, exercising a pre- 
cisely corresponding pressure. Under these circumstances the 
form assumed by the sides of our figure will be that of a portion 
of an unduloid. If we increase the pressure between the discs, 
the peripheral surface of oil will bulge out more and more, and 
will presently constitute a portion of a sphere. But we may 
continue the process yet further, and within certain limits we shall 
find that the system remains perfectly stable. What is this new 
curved surface which has arisen out of the sphere, as the latter 
was produced from the unduloid? It is no other than a portion 
of a nodoid, that part which in Fig. 66 lies between such limits as 
M and Nn. But this surface, which is concave in both directions 
towards the surface of the oil within, is exerting a pressure upon 
the latter, just as did the sphere out of which a moment ago it 
was transformed; and we had just stated, in considering the 
previous experiment, that the pressure inwards exerted by the 
nodoid was a negative one. The explanation of this seeming 
discrepancy lies in the simple fact that, if we follow the outline 


‘ 


Fig. 66. 


v] OF FIGURES OF EQUILIBRIUM 22: 


Cl 


of our nodoid curve in Fig. 66 from o, p, the surface concerned 
in the former case, to M, N, that concerned in the present, we shall 
see that in the two experiments the surface of the liquid is not 
homologous, but lies on the positive side of the curve in the one 
case and on the negative side in the other. 


Of all the surfaces which we have been describing, the sphere 
is the only one which can enclose space; the others can only help 
to do so, in combination with one another or with the sphere itself. 
Thus we have seen that, in normal equilibrium, the cylindrical 
vesicle is closed at either end by a portion of a sphere, and so on. 
Moreover the sphere is not only the only one of our figures which 
can enclose a finite space; it is also, of all possible figures, that 
which encloses the greatest volume with the least area of surface ; 
it is strictly and absolutely the surface of minimal area, and it 
is therefore the form which will be naturally assumed by a uni- 
cellular organism (just as by a raindrop), when it is practically 
homogeneous and when, like Orbulina floating in the ocean, its 
surroundings are likewise practically homogeneous and sym- 
metrical. It is only relatively speaking that all the rest are 
surfaces minimae areae; they are so, that is to say, under the 
given conditions, which involve various forms of pressure or 
restraint. Such restraints are imposed, for instance, by the 
pipes or annuli with the help of which we draw out our cylindrical 
or unduloid oil-globule or soap-bubble; and in the case of the 
organic cell, similar restraints are constantly supplied by solidifica- 
tion, partial or complete, local or general, of the cell-wall. 

Before we pass to biological illustrations of our surface-tension 
figures, we have still another prelimimary matter to deal with. 
We have seen from our description of two of Plateau’s classical 
experiments, that at some particular point one type of surface 
gives place to another; and again, we know that, when we draw 
out our soap-bubble into and then beyond a cylinder, there comes 
a certain definite point at which our bubble breaks in two, and 
leaves us with two bubbles of which each is a sphere, or a portion 
of a sphere. In short there are certain definite limits to the 
dimensions of our figures, within which limits equilibrium is 
stable but at which it becomes unstable, and above which it 


T. G. 15s 


226 THE FORMS OF CELLS (cH. 


breaks down. Moreover in our composite surfaces, when the 
cylinder for instance is capped by two spherical cups or lenticular 
. discs, there is a well-defined ratio which regulates their respective 
curvatures, and therefore their respective dimensions. These two 
matters we may deal with together. 

Let us imagine a liquid drop which by appropriate conditions 
has been made to assume the form of a cylinder; we have already 
seen that its ends will be terminated by portions of spheres. 
Since one and the same liquid film covers the sides and ends of 
the drop (or since one.and the same delicate membrane encloses 
the sides and ends of the cell), we assume the surface-tension (7') 
to be everywhere identical; and it follows, since the internal 
fluid-pressure is also everywhere identical, that the expression 
(1/R + 1/R’) for the cylinder is equal to the corresponding expres- 
sion, which we may call (1/7 + 1/r’), in the case of the terminal 
spheres. Butin the cylinder 1/R’ = 0,and inthesphere 1/r = 1/r’. 
Therefore our relation of equality becomes 1/R = 2/r, or r = 2R; 
that is to say, the sphere in question has just twice the radius of 
the cylinder of which it forms a cap. 

And if Ob, the radius of the sphere, be equal to twice the radius 
(Oa) of the cylinder, it follows that the angle aOb is an angle of 

; 60°, and bOc is also an angle of 60°; 
that is to say, the are be is equal to 
ia. In other words, the spherical 
qi b — dise which (under the given conditions) 
caps our cylinder, is not a portion 
taken at haphazard, but is neither 
more nor less than that portion of a 
sphere which is subtended by a cone 
of 60°. Moreover, it is plain that 
the height of the spherical cap, de, 


— Ob — ab = R (2 — /3) = 0-27R, 


dl 


€ 


where R is the radius of our cylinder, 
or one-half the radius of our spherical 
cap: in other words the normal height of the spherical cap over 
the end of the cylindrical cell is just a very little more than one- 
eighth of the diameter of the cylinder, or of the radius of the 


Fig. 67. 


Vv] OF FIGURES OF EQUILIBRIUM 227 


sphere. And these are the proportions which we recognise, under 
normal circumstances, in such a case as the cylindrical cell of 
Spirogyra where its free end is capped by a portion of a sphere. 


Among the many important theoretical discoveries which we 
owe to Plateau, one to which we have just referred is of peculiar 
importance: namely that, with the exception of the sphere and 
the plane, the surfaces with which we have been dealing are only 
in complete equilibrium within certain dimensional limits, or in 
other words, have a certain definite limit of stability ; only the plane 
and the sphere, or any portions of a sphere, are perfectly stable, 
because they are perfectly symmetrical, figures. For experimental 
demonstration, the case of the cylinder is the simplest. If we 
produce a liquid film having the form of a cylinder, either by 


2 a ee eo erat 
A 


ee a ee | 


drawing out a bubble or by supporting between two rings a 
globule of oil, the experiment proceeds easily until the length of 
the cylinder becomes just about three times as great as its diameter. 
But somewhere about this limit the cylinder alters its form; it 
begins to narrow at the waist, so passing into an unduloid, and 
the deformation progresses quickly until at last our cylinder 
breaks in two, and its two halves assume a spherical form. It is 
found, by theoretical considerations, that the precise limit of 
stability is at the point when the length of the cylinder is exactly 
equal to its circumference, that is to say, when L = 27R, or when 
the ratio of length to diameter is represented by z. 

In the case of the catenoid, Plateau’s experimental procedure 
was as follows. To support his globule of oil (in, as usual, a 
mixture of alcohol and water of its own specific gravity), he used 

15—2 


228 THE FORMS OF CELLS [CH. 


a pair of metal rings, which happened to have a diameter of 
71 millimetres; and, in a series of experiments, he set these rings 
apart at distances of 55, 49, 47, 45, and 43 mm. successively. 
In each case he began by bringing his oil-globule into a cylindrical 
form, by sucking superfluous oil out of the drop until this result 
was attained; and always, for the reason with which we are now 
acquainted, the cylindrical sides were associated with spherical 
ends to the cylinder. On continuing to withdraw oil in the hope 
of converting these spherical ends into planes, he found, naturally, 
that the sides of the cylinder drew in to form a concave surface ; 
but it was by no means easy to get the extremities actually plane: 
and unless they were so, thus indicating that the surface-pressure 
of the drop was nil, the curvature of the sides could not be that 
of a catenoid. For in the first experiment, when the rings were 
55 mm. apart, as soon as the convexity of the ends was to a certain 
extent diminished, it spontaneously increased again; and the 
transverse constriction of the globule correspondingly deepened, 
until at a certain point equilibrium set in anew. Indeed, the more 
oil he removed, the more convex became the ends, until at last 
the increasing transverse constriction led to the breaking of the- 
oil-globule into two. In the third experiment, when the rings 
were 47 mm. apart, it was easy to obtain end-surfaces that were 
actually plane, and they remained so even though more oil was 
withdrawn, the transverse constriction deepening accordingly. 
Only after a considerable amount of oil had been sucked up did 
the plane terminal surface become gradually cohvex, and presently 
the narrow waist, narrowing more and more, broke across in the 
usual way. Finally in the fifth experiment, where the rings were 
still nearer together, it was again possible to bring the ends of the 
oil-globule to a plane surface, as in the third and fourth experiments, 
and to keep this surface plane in spite of some continued with- 
drawal of oil. But very soon the ends became gradually concave, 
and the concavity deepened as more and more oil was withdrawn, 
until at a certain limit, the whole oil-globule broke up in general 
disruption. 

We learn from this that the limiting size of the catenoid was 
reached when the distance of the supporting rings was to their 
diameter as 47 to 71, or, as nearly as possible, as two to three; 


v] OF FIGURES OF EQUILIBRIUM 229 


and as a matter of fact it can be shewn that 2/3 is the true 
theoretical value. Above this limit of 2/3, the inevitable convexity 
of the end-surfaces shows that a positive pressure inwards is being 
exerted by the surface film, and this teaches us that the sides of 
the figure actually constitute not a catenoid but an unduloid, 
whose spontaneous changes tend to a form of greater stability. 
Below the 2/3 limit the catenoid surface is essentially unstable, 
and the form into which it passes under certain conditions of 
disturbance such as that of the excessive withdrawal of oil, is 
that of a nodoid (Fig. 65 A). 

The unduloid has certain peculiar properties as regards its 
limitations of stability. But as to these we need mention two 
facts only: (1) that when the unduloid, which we produce with 
our soap-bubble or our oil-globule, consists of the figure containing 
a complete constriction, it has somewhat wide limits of stability ; 
but (2) if it contain the swollen portion, then equilibrium is limited 
to the condition that the figure consists simply of one complete 
unduloid, that is to say that its ends are constituted by the 
narrowest portions, and its middle by the widest portion of the 
entire curve. The theoretical proof of this latter fact is difficult, 
but if we take the proof for granted, the fact will serve to throw 
light on what we have learned regarding the stability of the cylinder. 
For, when we remember that the meridional section of our unduloid 
is generated by the rolling of an ellipse upon a straight line in its 
own plane, we shall easily see that the length of the entire unduloid 
is equal to the circumference of the generating ellipse. As the 
unduloid becomes less and less sinuous in outline, it gradually 
approaches, and in time reaches, the form of a cylinder; and 
correspondingly, the ellipse which generated it has its foci more 
and more approximated until it passes into a circle. The cylinder 
of a length equal to the circumference of its generating circle is 
therefore precisely homologous to an unduloid whose length is 
equal to the circumference of its generating ellipse; and this is 
just what we recognise as constituting one complete segment of 
the unduloid. 


While the figures of equilibrium which are at the same time 
surfaces of revolution are only six in number, there is an infinite 


230 THE FORMS OF CELLS [CH. 


number of figures of equilibrium, that is to say of surfaces of 
constant mean curvature, which are not surfaces of revolution: 
and it can be shewn mathematically that any given contour can 
be occupied by a finite portion of some one such surface, in stable 
equilibrium. The experimental verification of this theorem lies in 
the simple fact (already noted) that however we may bend a wire 
into a closed curve, plane or not plane, we may always, under 
appropriate precautions, fill the entire area with an unbroken 
film. 

Of the regular figures of equilibrium, that is to say surfaces 
of constant mean curvature, apart from the surfaces of revolution 
which we have discussed, the helicoid spiral is the most interesting 
to the biologist. This is a helicoid generated by a straight line 
perpendicular to an axis, about which it turns at a uniform rate 
while at the same time it slides, also uniformly, along this same 
axis. At any point in this surface, the curvatures are equal and 
of opposite sign, and the sum of the curvatures is accordingly nil. 
Among what are called “ruled surfaces” (which we may describe 
as surfaces capable of being defined by a system of stretched 
strings), the plane and the helicoid are the only two whose mean 
curvature is null, while the cylinder is the only one whose curvature 
is finite and constant. As this simplest of helicoids corresponds, 
in three dimensions, to what in two dimensions is merely a plane 
(the latter being generated by the rotation of a straight line about 
an axis without the superadded gliding motion which generates 
the helicoid), so there are other and much more complicated 
helicoids which correspond to the sphere, the unduloid and the 
rest of our figures of revolution, the generating planes of these 
latter being supposed to wind spirally about an axis. In the case 
of the cylinder it is obvious that the resulting figure is indistinguish- 
able from the cylinder itself. In the case of the unduloid we 
obtain a grooved spiral, such as we may meet with in nature (for 
instance in Spirocheetes, Bodo gracilis, etc.), and which accordingly 
it is of interest to us to be able to recognise as a surface of minimal 
area or constant curvature. 

The foregoing considerations deal with a small part only 
of the theory of surface tension, or of capillarity: with that 
part, namely, which relates to the forms of surface which are 


v] OF FIGURES OF EQUILIBRIUM 231 


capable of subsisting in equilibrium under the action of that force, 
either of itself or subject to certain simple constraints. And as 
yet we have limited ourselves to the case of a single surface, or 
of a single drop or bubble, leaving to another occasion a discussion 
of the forms assumed when such drops or vesicles meet and com- 
bine together. In short, what we have said may help us to under- 
stand the form of a cell,—considered, as with certain limitations 
we may legitimately consider it, as a liquid drop or liquid vesicle ; 
the conformation of a tissue or cell-aggregate must be dealt with — 
in the light of another series of theoretical considerations. In 
both cases, we can do no more than touch upon the fringe of a 
large and difficult subject. There are many forms capable of 
realisation under surface tension, and many of them doubtless to 
be recognised among organisms, which we cannot touch upon in 
this elementary account. The subject is a very general one: it 
is, in its essence, more mathematical than physical; it is part of 
the mathematics of surfaces, and only comes into relation with 
surface tension, because this physical phenomenon illustrates and 
exemplifies, in a concrete way, most of the simple and symmetrical 
conditions with which the general mathematical theory is capable 
of dealing. And before we pass to illustrate by biological examples 
the physical phenomena which we have described, we must be 
careful to remember that the physical conditions which we have 
hitherto presupposed will never be wholly realised in the organic 
cell. Its substance will never be a perfect fluid, and hence 
equilibrium will be more or less slowly reached; its surface will 
seldom be perfectly homogeneous, and therefore equilibrium will 
(in the fluid condition) seldom be perfectly attained; it will very 
often, or generally, be the seat of other forces, symmetrical or 
unsymmetrical; and all these causes will more or less perturb the 
effects of surface tension acting by itself. But we shall find that, 
on the whole, these effects of surface tension though modified are 
not obliterated nor even masked; and accordingly the phenomena 
to which I have devoted the foregoing pages will be found 
manifestly recurring and repeating themselves among the pheno- 
mena of the organic cell. 


In a spider’s web we find exemplified several of the principles 


232 THE FORMS OF CELLS [CH. 


of surface tension which we have now explained. The thread is 
formed out of the fluid secretion of a gland, and issues from the 
body as a semi-fluid cylinder, that is to say in the form of a surface 
of equilibrium, the force of expulsion giving it its elongation and 
that of surface tension giving it its circular section. It is prevented, 
by almost immediate solidification on exposure to the air, from 
breaking up into separate drops or spherules, as it would otherwise 
tend to do as soon as the length of the cylinder had passed its 
limit of stability. But it is otherwise with the sticky secretion 
which, coming from another gland, is simultaneously poured over 
the issuing thread when it is to form the spiral portion of the 
web. This latter secretion is more fluid than the first, and retains 
its fluidity for a very much longer time, finally drying up after 
several hours. By capillarity it “wets” the thread, spreading 
itself over it in an even film, which film is now itself a cylinder. 
But this liquid cylinder has its limit of stability when its length 
equals its own circumference, and therefore just at the points so 
defined it tends to disrupt into separate segments: or rather, in 
the actual case, at points somewhat more distant, owing to the 
imperfect fluidity of the viscous film, and still more to the frictional 
drag upon it of the inner solid cylinder, or thread, with which it 
is in contact. The cylinder disrupts in the usual manner, passing 
first into the wavy outline of an unduloid, whose swollen portions 
swell more and more till the contracted parts break asunder, and 
we arrive at a series of spherical drops or beads, of equal size, 
strung at equal intervals along the thread. If we try to spread 
varnish over a thin stretched wire, we produce automatically the 
same identical result*; unless our varnish be such as to dry almost 
instantaneously, it gathers into beads, and do what we can, we 
fail to spread it smooth. It follows that, according to the viscidity 
and drying power of the varnish, the process may stop or seem to 
stop at any point short of the formation of the perfect spherules ; 
it is quite possible, therefore, that as our final stage we may only 
obtain half-formed beads, or the wavy outline of an unduloid. 
The formation of the beads may be facilitated or hastened by 
jerking the stretched thread, as the spider actually does: the 


* Felix Plateau recommends the use of a weighted thread, or plumb-line, 
drawn up out of a jar of water or oil; Phil. Mag. xxxtv, p. 246, 1867. 


v| OF SPIDERS’ WEBS 233 


effect of the jerk being to disturb and destroy the unstable 
equilibrium of the viscid cylinder*. Another very curious 
phenomenon here presents itself. 

In Plateau’s experimental separation of a cylinder of oil into 
two spherical portions, it was noticed that, when contact was 
nearly broken, that is to say when the narrow neck of the unduloid 
had become very thin, the two spherical bullae, instead of absorbing 
the fluid out of the narrow neck into themselves as they had done 
with the preceding portion, drew out this small remaining part of 
the liquid into a thin thread as they completed their spherical 
form and consequently receded from one another: the reason being 
that, after the thread or “neck” has reached a certain tenuity, 
the internal friction of the fluid prevents or retards its rapid exit 
from the little thread to the adjacent spherule. It is for the same 
reason that we are able to draw a glass rod or tube, which we have 
heated in the middle, into a long and uniform cylinder or thread, 
by quickly separating the two ends. But in the case of the glass 
~ rod, the long thin intermediate cylinder quickly cools and solidifies, 
while in the ordinary separation of a liquid cylinder the corre- 
sponding intermediate cylinder remains liquid; and therefore, like 
any other liquid cylinder, it is liable to break up, provided that its 
dimensions exceed the normal limit of stability. And its length 
is generally such that it breaks at two points, thus leaving two 
terminal portions continuous with the spheres and becoming 
confluent with these, and one median portion which resolves itself 
into a comparatively tiny spherical drop, midway between the 
original and larger two. Occasionally, the same process of forma- 
tion of a connecting thread repeats itself a second time, between 
the small intermediate spherule and the large spheres; and in this 
case we obviously obtain two additional spherules, still smaller in 
size, and lying one on either side of our first little one. This whole 
phenomenon, of equal and regularly interspaced beads, often with 
little beads regularly interspaced between the larger ones, and 
possibly also even a third series of still smaller beads regularly 
intercalated, may be easily observed in a spider’s web, such as 
that of Epeira, very often with beautiful regularity,—which 


* Cf. Boys, C. V., On Quartz Fibres, Nature, July 11, 1889; Warburton, C., 
The Spinning Apparatus of Geometric Spiders, Q.J.M.S. xxx1, pp. 29-39, 1890. 


234 THE FORMS OF CELLS [CH. 


naturally, however, is sometimes interrupted and disturbed owing 
to a slight want of homogeneity in the secreted fluid; and the 
same phenomenon is repeated on a grosser scale when the web is 
bespangled with dew, and every thread bestrung with pearls 
innumerable. To the older naturalists, these regularly arranged 
and beautifully formed globules on the spider’s web were a cause 
of great wonder and admiration. Blackwall, counting some 
twenty globules in a tenth of an inch, calculated that a large 
garden-spider’s web comprised about 120,000 globules; the net 
was spun and finished in about forty minutes, and Blackwall was 
evidently filled with astonishment at the skill and quickness with 
which the spider manufactured these little beads. And no wonder, 
for according to the above estimate they had to be made at the 
rate of about 50 per second*. E 

The little delicate beads which stud the long thin pseudopodia 
of a foraminifer, such as Gromia, or which in like manner appear 


See 


Fig. 69. Hair of Trianea, in glycerine. (After Berthold.) 


upon the cylindrical film of protoplasm which covers the long 
radiating spicules of Globigerina, represent an identical phenomenon. 
Indeed there are many cases, in which we may study in a proto- 
plasmic filament the whole process of formation of such beads. 
If we squeeze out on to’a slide the viscid contents of a mistletoe 
berry, the long sticky threads into which the substance runs shew 
the whole phenomenon particularly well. Another way to 
demonstrate it was noticed many years ago by Hofmeister and 
afterwards explained by Berthold. The hairs of certain water- 
plants, such as Hydrocharis or Trianea, constitute very long cylin- 
drical cells, the protoplasm being supported, and maintained in 
equilibrium by its contact with the cell-wall. But if we immerse 
the filament in some dense fluid, a little sugar-solution for instance, 
or dilute glycerine, the cell-sap tends to diffuse outwards, the proto- 
plasm parts company with its surrounding and supporting wall, 


* J. Blackwall, Spiders of Great Britain (Ray Society), 1859, p. 10; Trans. 
Linn. Soc. Xvi, p. 477, 1833. 


v] OF GLOBULES OR BEADS 235 


and lies free as a protoplasmic cylinder in the interior of the cell. 
Thereupon it immediately shews signs of instability, and commences 
to disrupt. It tends to gather into.spheres, which however, as in 
our illustration, may be prevented by their narrow quarters from 
assuming the complete spherical form; and in between these 
spheres, we have more or less regularly alternate ones, of smaller 
size*. Similar, but less regular, beads or droplets may be caused to 
appear, under stimulation by an alternating current, in the proto- 
plasmic threads within the living cells of the hairs of Tradescantia. 
The explanation usually given is, that the viscosity of the proto- 


Fig. 70. Phases of a Splash. (From Worthington.) 


plasm is reduced, or its fluidity increased; but an increase of the 
surface tension would seem a more likely reason f. 


We may take note here of a remarkable series of phenomena, 
which, though they seem at first sight to be of a very different 
order, are closely related to the phenomena which attend and 
which bring about the breaking-up of a liquid cylinder or thread. 

In some of Mr Worthington’s most beautiful ‘experiments on 

* The intermediate spherules appear, with great regularity and beauty, whenever 
a liquid jet breaks up into drops; see the instantaneous photographs in Poynting 


and Thomson’s Properties of Matter, pp. 151, 152, (ed. 1907). 
7 Kihne, Untersuchungen iiber das Protoplasma, 1864, p. 75, etc. 


236 THE FORMS OF CELLS [cH. 


splashes, it was found that the fall of a round pebble into water 
from a considerable height, caused the rise of a filmy sheet of water 
in the form of a cup or cylinder; and the edge of this cylindrical 
film tended to be cut up into alternate lobes and notches, and the 
prominent lobes or “jets” tended, in more extreme cases, to break 
off or to break up into spherical beads (Fig. 70)*. <A. precisely 
similar appearance is seen, on a great scale, in the thin edge of a 
breaking wave: when the smooth cylindrical edge, at a given 
moment, shoots out an array of tiny jets which break up into 
the droplets which constitute ‘‘spray’”’ (Fig. 71, a, 6). We 
are at once reminded of the beautifully symmetrical notching on 
the calycles of many hydroids, which little cups before they became 
stiff and rigid had begun their existence as liquid or semi-liquid 
films. 


b 
Fig. 71. A breaking wave. (From Worthington.) 


The phenomenon is two-fold. In the first place, the edge of 
our tubular or crater-like film forms a liquid ring or annulus, 
which is closely comparable with the liquid thread or cylinder 
which we have just been considering, if only we conceive the thread 
to be bent round into the ring. And accordingly, just as the thread 
spontaneously segments, first into an unduloid, and then into 
separate spherical drops, so likewise will the edge of our annulus 
tend to do. This phase of notching, or beading, of the edge of 
the film is beautifully seen in many of Worthington’s experiments f. 
In the second place, the very fact of the rising of the crater means 
that liquid is flowing up from below towards the rim; and the 
segmentation of the rim means that channels of easier flow are 


* A Study of Splashes, 1908, p. 38, etc.; Segmentation of a Liquid Annulus, 
Proc. Roy. Soc. xxx, pp. 49-60, 1880. 

7 Cf. ibid. pp. 17, 77. The same phenomenon is beautifully and continuously 
evident when a strong jet of water from a tap impinges on a curved surface and then 
shoots off it. 


v] THE SHAPE OF A SPLASH 237 


created, along which the liquid is led, or is driven, into the pro- 
tuberances: and these are thus exaggerated into the jets or arms 
which are sometimes so conspicuous at the edge of the crater. 
In short, any film or film-like cup, fluid or semi-fluid in its consis- 
tency, will, like the straight liquid cylinder, be unstable: and its 
instability will manifest itself (among other ways) in a tendency 
to segmentation or notching of the edge; and just such a peripheral 
notching is a conspicuous feature of many minute organic cup-like 
structures. In the case of the hydroid calycle (Fig. 72), we are led 
to the conclusion that the two common and conspicuous features 
of notching or indentation of the cup, and of constriction or 
annulation of the long cylindrical stem, are phenomena of the 
same order and are due to surface-tension in both cases alike. 


Fig. 72. Calycles of Campanularian zoophytes. (A) C. integra; 
(B) C. groenlandica; (C) C. bispinosa; (D) C. raridentata. 


Another phenomenon displayed in the same experiments is the 
formation of a rope-like or cord-like thickening of the edge of the 
annulus. This is due to the more or less sudden checking at the 
rim of the flow of liquid rising from below: and a similar peripheral 
thickening is frequently seen, not only in some of our hydroid 
cups, but in many Vorticellas (cf. Fig. 75), and other organic 
cup-like conformations. A perusal of Mr Worthington’s book 
will soon suggest that these are not the only manifestations of 
surface-tension in connection with splashes which present curious 
resemblances and analogies to phenomena of organic form. 

The phenomena of an ordinary liquid splash are so swiftly 


238 THE FORMS OF CELLS [CH. 


transitory that their study is only rendered possible by “instan- 
taneous” photography: but this excessive rapidity is not an 
essential part of the phenomenon. For instance, we can repeat 
and demonstrate many of the simpler phenomena, in a permanent 
or quasi-permanent form, by splashing water on to a surface of 
dry sand, or by firing a bullet into a soft metal target. There is 
nothing, then, to prevent a slow and lasting manifestation, in 
a viscous medium such as a protoplasmic organism, of phenomena 
which appear and disappear with prodigious rapidity in a more 
mobile liquid. Nor is there anything peculiar in the “splash” 
itself; it is simply a convenient method of setting up certain 
motions or currents, and producing certain surface-forms, in a 
liquid medium,—or even in such an extremely imperfect fluid as 
is represented (in another series of experiments) by a bed of sand. 
Accordingly, we have a large range of possible conditions under 
which the organism might conceivably display configurations 
analogous to, or identical with, those which Mr Worthington has 
shewn us how to exhibit by one particular experimental method. 

To one who has watched the potter at his wheel, it is plain 
that the potter's thumb, like the glass-blower’s blast of air, 
depends for its efficacy upon the physical properties of the 
medium on which it operates, which for the time being is essentially 
a fluid. The cup and the saucer, like the tube and the bulb, 
display (in their simple and primitive forms) beautiful surfaces of 
equilibrium as manifested under certain limiting conditions. 
They are neither more nor less than glorified “splashes,”’ formed 
slowly, under conditions of restraint which enhance or reveal 
their mathematical symmetry. We have seen, and we shall see 
again before we are done, that the art of the glass-blower is full 
of lessons for the naturalist as also for the physicist: illustrating 
as it does the development of a host of mathematical configura- 
tions and organic conformations which depend essentially on the 
establishment of a constant and uniform pressure within a closed 
elastic shell or fluid envelope. In like manner the potter’s art 
illustrates the somewhat obscurer and more complex problems 
(scarcely less frequent in biology) of a figure of equilibrium which 
is an open surface, or solid, of revolution. It is clear, at the same 
time, that the two series of problems are closely akin; for the 


v| THE SHAPE OF A SPLASH 239 


glass-blower can make most things that the potter makes, by 
cutting off portions of his hollow ware. And besides, when this 
fails, and the glass-blower, ceasing to blow, begins to use his rod 
to trim the sides or turn the edges of wineglass or of beaker, he 
is merely borrowing a trick from the craft of the potter. 

It would be venturesome indeed to extend our comparison 
with these liquid surface-tension phenomena from the cup or 
calycle of the hydrozoon to the little hydroid polype within: and 
yet I feel convinced that there is something to be learned by such 
a comparison, though not without much detailed consideration 
and mathematical study of the surfaces concerned. The cylin- 
drical body of the tiny polype, the jet-like row of tentacles, the 
beaded annulations which these tentacles exhibit, the web-like 
film which sometimes (when they stand a little way apart) conjoins 
their bases, the thin annular film of tissue which surrounds the 
little organism’s mouth, and the manner in which this annular 
“peristome” contracts*, like a shrinking soap-bubble, to close the 
aperture, are every one of them features to which we may find 
a singular and striking parallel in the surface-tension phenomena 
which Mr Worthington has illustrated and demonstrated in the 
case of the splash. 

Here however, we may freely confess that we are for the 
present on the uncertain ground of suggestion and conjecture; 
and so must we remain, in regard to many other simple and 
symmetrical organic forms, until their form and dynamical 
stability shall have been investigated by the mathematician: in 
other words, until the mathematicians shall have become persuaded 
that there is an immense unworked field wherein they may labour, 
in the detailed study of organic form. 


According to Plateau, the viscidity of the liquid, while it 
helps to retard the breaking up of the cylinder and so increases 
the length of the segments beyond that which theory demands, 
has nevertheless less influence in this direction than we might 
have expected. On the other hand, any external support or 
adhesion, such as contact with a solid body, will be equivalent to 
a reduction of surface-tension and so will very greatly increase the 


* See a Study of Splashes, p. 54. 


240 THE FORMS OF CELLS [cx. 


stability of our cylinder. It is for this reason that the mercury 
in our thermometer tubes does not as a rule separate into drops, 
though it occasionally does so, much to our inconvenience. And 
again it is for this reason that the protoplasm in a long and growing 
tubular or cylindrical cell does not necessarily divide into separate 
cells and internodes, until the length of these far exceeds the 
theoretic limits. Of course however and whenever it does so, we 
must, without ever excluding the agency of surface tension, 
remember that there may be other forces affecting the latter, and 
accelerating or retarding that manifestation of surface tension by 
which the cell is actually rounded off and divided. 

In most liquids, Plateau asserts that, on the average, the 
influence of viscosity is such as to cause the cylinder to segment 
when its length is about four times, or at most from four to six 
times that of its diameter: instead of a fraction over three times 
as, in a perfect fluid, theory would demand. If we take it at 
four times, it may then be shewn that the resulting spheres would 
have a diameter of about 1-8 times, and their distance apart would 
be equal to about 2-2 times the diameter of the original cylinder. 
The calculation is not difficult which would shew how these 
numbers are altered in the case of a cylinder formed around a solid 
core, as in the case of the spider’s web. Plateau has also made 
the interesting observation that the time taken in the process of 
division of the cylinder is directly proportional to the diameter 
of the cylinder, while varying considerably with the nature of the 
liquid. This question, of the time occupied in the division of a 
cell or filament, in relation to the dimensions of the latter, has not 
so far as I know been enquired into by biologists. 


From the simple fact that the sphere is of all surfaces that 
whose surface-area for a given volume is an absolute minimum, 
we have already seen it to be plain that it is the one and only 
figure of equilibrium which will be assumed under surface-tension 
by a drop or vesicle, when no other disturbing factors are present. 
One of the most important of these disturbing factors will be 
introduced, in the form of complicated tensions and pressures, 
when one drop is in contact with another drop and when a system 
of intermediate films or partition walls is developed between them. 


vl OF ASYMMETRY AND ANISOTROPY 241 


This subject we shall discuss later, in connection with cell- 
aggregates or tissues, and we shall find that further theoretical 
considerations are needed as a preliminary to any such enquiry. 
Meanwhile let us consider a few cases of the forms of cells, either 
solitary, or in such simple aggregates that their individual form is 
little disturbed thereby. 

Let us clearly understand that the cases we are about to 
consider are those cases where the perfect symmetry of the sphere 
is replaced by another symmetry, less complete, such as that of 
an ellipsoidal or cylindrical cell. The cases of asymmetrical 
deformation or displacement, such as is illustrated in the production 
of a bud or the development of a lateral branch, are much simpler. 
For here we need only assume a slight. and localised variation of 
surface-tension, such as may be brought about in various ways 
through the heterogeneous chemistry of the cell; to this point 
we shall return in our chapter on Adsorption. But the diffused 
and graded asymmetry of the system, which brings about for 
instance the ellipsoidal shape of a yeast-cell, is another matter. 

If the sphere be the one surface of complete symmetry and 
therefore of independent equilibrium, it follows that in every cell 
which is otherwise conformed there must be some definite force 
to cause its departure from sphericity; and if this cause be the 
very simple and obvious one of the resistance offered by a solidified 
envelope, such as an egg-shell or firm cell-wall, we must still seek 
for the deforming force which was in action to bring about the 
given shape, prior to the assumption of rigidity. Such a cause 
may be either external to, or may lie within, the cell itself. On 
the one hand it may be due to external pressure or to some form 
of mechanical restraint: as it is in all our experiments in which 
we submit our bubble to the partial restraint of discs or rings or 
more complicated cages of wire; and on the other hand it may be 
due to intrinsic causes, which must come under the head either of 
differences of internal pressure, or of lack of homogeneity or 
isotropy in the surface itself*. 


* A case which we have not specially considered, but which may be found to 
deserve consideration in biology, is that of a cell or drop suspended in a liquid of 
varying density, for instance in the upper layers of a fluid (e.g. sea-water) at whose 
surface condensation is going on, so as to produce a steady density-gradient. In 
this case the normally spherical drop will be flattened into an oval form, with its 


GS (Se 16 


242 THE FORMS OF CELLS [CH. 


Our full formula of equilibrium, or equation to an elastic 
surface, is P= p,+(T/R+T'/R’), where P is the internal 
pressure, p, any extraneous pressure normal to the surface, R, R’ 
the radii of curvature at a point, and 7, T’, the corresponding 
tensions, normal to one another, of the envelope. 

Now in any given form which we are seeking to account for, 
R, R’ are known quantities; but all the other factors of the equation 
are unknown and subject to enquiry. And somehow or other, by 
this formula, we must account for the form of any solitary cell 
whatsoever (provided always that it be not formed by successive 
stages of solidification), the cylindrical cell of Spirogyra, the 
ellipsoidal yeast-cell, or (as we shall see in another chapter) the 
shape of the egg of any bird. In using this formula hitherto, we 
have taken it in a simplified form, that is to say we have made 
several limiting assumptions. We have assumed that P was 
simply the uniform hydrostatic pressure, equal in all directions, 
of a body of liquid; we have assumed that the tension 7 was 
simply due to surface-tension in a homogeneous liquid film, and 
was therefore equal in all directions, so that T = T’; and we have 
only dealt with surfaces, or parts of a surface, where extraneous 
pressure, p,,, Was non-existent. Now in the case of a bird’s egg, 
the external pressure p,, that is to say the pressure exercised by 
the walls of the oviduct, will be found to be a very important 
factor; but in the case of the yeast-cell or the Spirogyra, wholly 
immersed in water, no such external pressure comes into play. 
We are accordingly left, in such cases as these last, with two 
hypotheses, namely that the departure from a spherical form is due 
to inequalities in the internal pressure P, or else to inequalities in 
the tension 7’, that is to say to a difference between 7 and 7”. 
In other words, it is theoretically possible that the oval form of 
a yeast-cell is due to a greater internal pressure, a greater 
“tendency to grow,” in the direction of the longer axis of the 
ellipse, or alternatively, that with equal and symmetrical tendencies 
to growth there is associated a difference of external resistance in 


maximum surface-curvature lying at the level where the densities of the drop 
and the surrounding liquid are just equal. The sectional outline of the drop has 
been shewn to be not a true oval or ellipse, but a somewhat complicated quartic 
curve. (Rice, Phil. Mag. Jan. 1915.) 


v] OF ASYMMETRY AND ANISOTROPY 243 


respect of the tension of the cell-wall. Now the former hypothesis 
is not impossible; the protoplasm is far from being a perfect fluid ; 
it is the seat of various internal forces, sometimes manifestly 
polar; and accordingly it is quite possible that the internal 
forces, osmotic and other, which lead to an increase of the content 
of the cell and are manifested in pressure outwardly directed 
upon its wall may be unsymmetrical, and such as to lead to a 
deformation of what would otherwise be a simple sphere. But 
while this hypothesis is not impossible, it is not very easy of 
acceptance. The protoplasm, though not a perfect fluid, has yet 
on the whole the properties of a fluid; within the small compass 
of the cell there is little room for the development of unsymmetrical 
pressures; and, in such a case as Spirogyra, where a large part of 
the cavity is filled by a fluid and watery cell-sap, the conditions 
are still more obviously those under which a uniform hydrostatic 
pressure is to be expected. But in variations of 7, that is to say 
of the specific surface-tension per unit area, we have an ample 
field for all the various deformations with which we shall have to 
deal. Our condition now is, that (7/R + T’/R’) = a constant; but 
it no longer follows, though it may still often be the case, that this 
will represent a surface of absolute minimal area. As soon as T 
and T” become unequal, it is obvious that we are no longer dealing 
with a perfectly liquid surface film; but its departure from a 
perfect fluidity may be of all degrees, from that of a slight non- 
isotropic viscosity to the state of a firm elastic membrane*. And 
it matters little whether this viscosity or semi-rigidity be mani- 
fested in the self-same layer which is still a part of the protoplasm 
of the cell, or in a layer which is completely differentiated into a 
distinct and separate membrane. As soon as, by secretion or 
“adsorption,” the molecular constitution of the surface layer is 
altered, itis clearly conceivable that the alteration, or the secondary 
chemical changes which follow it, may be such as to produce an 
anisotropy, and to render the molecular forces less capable in 
one direction than another of exerting that contractile force by 
which they are striving to reduce to an absolute minimum the 


* Indeed any non-isotropic stiffness, even though T remained uniform, would 
simulate, and be indistinguishable from, a condition of non-stiffness and non- 
isotropic 7’. 


16—2 


244 THE FORMS OF CELLS [CH. 


surface area of the cell. A slight imequality in two opposite 
directions will produce the ellipsoid cell, and a very great in- 
equality will give rise to the cylindrical cell*. 

I take it therefore, that the cylindrical cell of Spirogyra, or 
any other cylindrical cell which grows in freedom from any 
manifest external restraint, has assumed that particular form 
simply by reason of the molecular constitution of its developing 
surface-membrane; and that this molecular constitution was 
anisotropous, in such a way as to render extension easier in one 
direction than another. 

Such a lack of homogeneity or of isotropy, in the cell-wall is 
often rendered visible, especially in plant-cells, in various ways, 
in the form of concentric lamellae, annular and spiral striations, 
and the like. 

But this phenomenon, while it brings about a certain departure 
from complete symmetry, is still compatible with, and coexistent 
with, many of the phenomena which we have seen to be associated 
with surface-tension. The symmetry of tensions still leaves the 
cell a solid of revolution, and its surface is still a surface of equi- 
librium. The fluid pressure within the cylinder still causes the 
film or membrane which caps its ends to be of a spherical form. 
And in the young cell, where the surface pellicle is absent or but 
little differentiated, as for instance in the odgonium of Achlya, 
or in the young zygospore of Spirogyra, we always see the tendency 
of the entire structure towards a spherical form reasserting itself: 
unless, as in the latter case, it be overcome by direct compression 
within the cylindrical mother-cell. Moreover, in those cases 
where the adult filament consists of cylindrical cells, we see that 
the young, germinating spore, at first spherical, very soon assumes 
with growth an elliptical or ovoid form: the direct result of an 
incipient anisotropy of its envelope, which when more developed 
will convert the ovoid into a cylinder. We may also notice that 
a truly cylindrical cell is comparatively rare; for in most cases, 
what we call a cylindrical cell shews a distinct bulging of its sides; 
it is not truly a cylinder, but a portion of a spheroid or ellipsoid. 


* A non-symmetry of 7’ and 7” might also be capable of explanation as a result 
of “liquid crystallisation.” This hypothesis is referred to, in connection with the 
blood-corpuscles, on p. 272. 


v] OF ASYMMETRY AND ANISOTROPY 245 


Unicellular organisms in general, including the protozoa, the 
unicellular cryptogams, the various bacteria, and the free, 
isolated cells, spores, ova, etc. of higher organisms, are referable 
for the most part to a very small number of typical forms; but 
besides a certain number of others which may be so referable, 
though obscurely, there are obviously many others in which 
either no symmetry is to be recognized, or in which the form is 
clearly not one of equilibrium. Among these latter we have 
Amoeba itself, and all manner of amoeboid organisms, and also 
many curiously shaped cells, such as the Trypanosomes and various 
other aberrant Infusoria. We shall return to the consideration of 
these; but in the meanwhile it will suffice to say that, as their 
surfaces are not equilibrium-surfaces, so neither are the living 
cells themselves in any stable equilibrium. On the contrary, they 
are in continual flux and movement, each portion of the surface 
constantly changing its form, and passing from one phase to 
another of an equilibrium which is never stable for more than 
a moment. The former class, which rest in stable equilibrium, 
must fall (as we have seen) into two classes,—those whose equi- 
librium arises from liquid surface-tension alone, and those in 
whose conformation some other pressure or restraint has been 
superimposed upon ordinary surface-tension. 

To the fact that these little organisms belong to an order of 
magnitude in which form is mainly, if not wholly, conditioned and 
controlled by molecular forces, is due the limited range of 
forms which they actually exhibit. These forms vary according 
to varying physical conditions. Sometimes they do so in so regular 
and orderly a way that we instinctively explain them merely as 
“phases of a life-history,” and leave physical properties and 
physical causation alone: but many of their variations of form we 
treat as exceptional, abnormal, decadent or morbid, and are apt 
to pass these over in neglect, while we give our attention to what 
we suppose to be the typical or “characteristic” form or attitude. 
In the case of the smallest organisms, the bacteria, micrococci, 
and so forth, the range of form is especially limited, owing to their 
minuteness, the powerful pressure which their highly curved 
surfaces exert, and the comparatively homogeneous nature of their 
substance. But within their narrow range of possible diversity 


246 THE FORMS OF CELLS [CH. 


these minute organisms are protean in their changes of form. 
A certain species will not only change its shape from stage to 
stage of its little “cycle” of life; but it will be remarkably different 
in outward form according to the circumstances under which we 
find it, or the histological treatment to which we submit it. Hence 
the pathological student, commencing the study of bacteriology, 
is early warned to pay little heed to differences of form, for purposes 
of recognition or specific identification. Whatever grounds we 
may have for attributing to these organisms a permanent or stable 
specific identity (after the fashion of the higher plants and animals), 
we can seldom safely do so on the ground of definite and always 
recognisable form: we may often be inclined, in short, to ascribe 


Fig. 73. A flagellate “monad,” Distigma Fig. 74. Noctiluca miliaris. 
proteus, Ehr. (After Saville Kent.) 


to them a physiological (sometimes a “ pathogenic’), rather than 
a morphological specificity. 


Among the Infusoria, we have a small number of forms whose 
symmetry is distinctly spherical, for instance among the small 
flagellate monads; but even these are seldom actually spherical 
except when we see them in a non-flagellate and more or less 
encysted or “resting” stage. In this condition, it need hardly be 
remarked that the spherical form is common and general among 
a great variety of unicellular organisms. When our little monad 
developes a flagellum, that is in itself an indication of “polarity” 
or symmetrical non-homogeneity of the cell; and accordingly, we 


v] OF VARIOUS UNDULOIDS 247 


usually see signs of an unequal tension of the membrane in the 
neighbourhood of the base of the flagellum. Here the tension is 
usually less than elsewhere, and the radius of curvature is accord- 
ingly less: in other words that end of the cell is drawn out to a 
tapering point (Fig. 73). But sometimes it is the other way, as 
in Noctiluca, where the large flagellum springs from a depression 
in the otherwise uniformly rounded cell. In this case the explan- 
ation seems to lie in the many strands of radiating protoplasm 
which converge upon this point, and may be supposed to keep it 
relatively fixed by their viscosity, while the rest of the cell-surface 
is free to expand (Fig. 74). 

A very large number of Infusoria represent unduloids, or 
portions of unduloids, and this type of surface appears and 
reappears in a great variety of forms. The cups of the various 
species of Vorticella (Fig. 75) are nothing in the world but a 


VUYUUV 90S 


1g. od 


Fig. 75. Various species of Vorticella. (Mostly after Saville Kent.) 


beautiful series of unduloids, or partial unduloids, in every grada- 
tion from a form that is all but cylindrical to one that is all but 
a perfect sphere. These unduloids are not completely symmetrical, 
but they are such unduloids as develop themselves when we 
suspend an oil-globule between two unequal rings, or blow a 
soap-bubble between two unequal pipes; for, just as in these 
. cases, the surface of our Vorticella bell finds its terminal supports, 
on the one hand in its attachment to its narrow stalk, and on the 
other in the thickened ring from which spring its circumoral cilia. 
And here let me say, that a point or zone from which cilia arise 
would seem always to have a peculiar relation to the surrounding 
tensions. It usually forms a sharp salient, a prominent point 
or ridge, as in our little monads of Fig. 73; shewing that, 
in its formation, the surface tension had here locally diminished. 
But if such a ridge or fillet consolidate in the least degree, it 
becomes a source of strength, and a point d’appui for the adjacent 
film. We shall deal with this point again in the next chapter. 


248 THE FORMS OF CELLS [CH. 


Precisely the same series of unduloid forms may be traced in 
even greater variety among various other families or genera of the 


Fig. 77. Various species of Tintinnus, Dinobryon and Codonella. (After 
Saville Kent and others.) 


Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen 
merely in the contour of the soft semifluid body of the living 
animal. At other times, as in Salpingoeca, Tin- 
tinnus, and many other genera, we have a distinct. _ 
membranous cup, separate from the animal, but 
originally secreted by, and moulded upon, its 
semifluid living surface. Here we have an excellent 
illustration of the contrast between the different 
ways in which such a structure may be regarded 
and interpreted. The teleological explanation is 
that it is developed for the sake of protection, as a 
domicile and shelter for the little organism within. 
The mechanical explanation of the physicist (seeking 
only after the “efficient,’”’ and not the “final” cause), is that it is 


Fig. 78. 
Vaginicola. 


v] OF VARIOUS UNDULOIDS 249 


present, and has its actual conformation, by reason of certain 
chemico-physical conditions: that it was inevitable, under the 
given conditions, that certain constituent 
substances actually present in the proto- | 
plasm should be aggregated by molecular 
forces in its surface layer; that under this 
adsorptive process, the conditions con- 
tinuing favourable, the particles should _ , 
accumulate and concentrate till they \ 
formed (with the help of the surrounding  ‘ 
medium) a pellicle or membrane, thicker 
or thinner as the case might be; that this 
surface pellicle or membrane was inevitably bound, by molecular 
forces, to become a surface of the least possible area which the 
circumstances permitted; that in the 
present case, the symmetry and “freedom” 
of the system permitted, and ipso facto 
caused, this surface to be a surface of 
revolution; and that of the few surfaces 
of revolution which, as being also surfaces 
minimae areae, were available, the undu- 
loid was manifestly the one permitted, 
and ipso facto caused, by the dimensions 
of the organisms and other circumstances 
of the case. And just as the thickness or 
thinness of the pellicle was obviously a 
subordinate matter, a mere matter of 
degree, so we also see that the actual 
outline of this or that particular unduloid 
is also a very subordinate matter, such as 
physico-chemical variants of a minute kind 
would suffice to bring about; for between 
the various unduloids which the various 
species of Vorticella represent, there is no 
more real difference than that difference li 
of ratio or degree which exists between Fig. 80. Trachelophyllum. 
two circles of different diameter, or two (After Wreszniowski.) 
lines of unequal length. 


Fig. 79.. Folliculina. 


250 THE FORMS OF CELLS [CH. 


In very many cases (of which Fig. 80 is an example), we have 
an unduloid form exhibited, not by a surrounding pellicle or shell, 
but by the soft, protoplasmic body of a ciliated organism. In 
such cases the form is mobile, and continually changes from one 
to another unduloid contour, according to the movements of the 
animal. We have here, apparently, to deal with an unstable 
equilibrium, and also sometimes with the more complicated 
problem of “stream-lines,” as in the difficult problems suggested 
by the form of a fish. But this whole class of cases, and of 
problems, we can merely take note of in passing, for their treat- 
ment is too hard for us. 


In considering such series of forms as the various unduloids 
which we have just been regarding, we are brought sharply 
- up (as in the case of our Bacteria or Micrococci) against the bio- 
logical concept of organic species. In the intense classificatory 
activity of the last hundred years, it has come about that every 
form which is apparently characteristic, that is to say which is 
capable of being described or portrayed, and capable of being 
recognised when met with again, has been recorded as a species,— 
for we need not concern ourselves with the occasional discussions, 
or individual opinions, as to whether such and such a form deserve 
“specific rank,” or be “only a variety.” And this secular labour 
is pursued in direct obedience to the precept of the Systema 
Naturae,—“ut sic in summa confusione rerum apparenti, summus 
conspiciatur Natwrae ordo.”’ In lke manner the physicist records, 
and is entitled to record, his many hundred “species” of snow- 
‘ erystals*, or of crystals of calcium carbonate. But regarding 
these latter species, the physicist makes no assumptions: he 
records them s¢mpliciter, as specific “forms”; he notes, as best 
he can, the circumstances, (such as temperature or humidity) 
under which they occur, in the hope of elucidating the conditions 
_ determining their formation; but above all, he does not introduce | 


* The case of the snow-crystals is a particularly interesting one; for their 
. “distribution” is in some ways analogous to what we find, for instance, among our 
microscopic skeletons of Radiolarians. That is to say, we may one day meet 
with myriads of some one particular form or species only, and another day with 
myriads of another; while at another time and place we may find species inter- 
mingled in inexhaustible variety. (Cf. e.g. J. Glaisher, [1]. London News, Feb. 17, 
1855; Q.J.M.S. m1, pp. 179-185, 1855). 


v] - OF FORM AND SPECIES 251 


the element of time, and of succession, or discuss their origin and 
affiliation as an historical sequence of events. But in biology, the 
term species carries with it many large, though often vague 
assumptions. Though the doctrine or concept of the “ permanence 
" of species” is dead and gone, yet a certain definite value, or sort 
of quasi-permanency, is still connoted by the term. Thus if a tiny 
foraminiferal shell, a Lagena for instance, be found living to-day, 
and a shell indistinguishable from it to the eye be found fossil 
in the Chalk or some other remote geological formation, the 
assumption is deemed legitimate that that species has “survived,” 
and has handed down its minute specific character or characters, 
from generation to generation, unchanged for untold myriads of 
years*. Or if the ancient forms be like to, rather than identical 
with the recent, we still assume an unbroken descent, accompanied 
by the hereditary transmission of common characters and pro- 
gressive variations. And if two identical forms be discovered at 
the ends of the earth, still (with occasional shght reservations on 
the score of possible “homoplasy”’), we build hypotheses on this 
fact of identity, taking it for granted that the two appertain to 
a common stock, whose dispersal in space must somehow be 
accounted for, its route traced, its epoch determined, and its 
causes discussed or discovered. In short, the naturalist admits 
no exception to the rule that a “natural classification” can only 
be a genealogical one, nor ever doubts that “The fact that we are 
able to classify organisms at all in accordance with the structural 
characteristics which they present, 1s due to the fact of the being 
related by descent}. But this great generalisation is apt in my 
opinion, to carry us too far. It may be safe and sure and helpful 
and illuminating when we apply it to such complex entities,— 
such thousand-fold resultants of the combination and permutation 
of many variable characters,—as a horse, a lon or an eagle; 
but (to my mind) it has a very different look, and a far less firm 
foundation, when we attempt to extend it to minute organisms 
whose specific characters are few and simple, whose simplicity 


* Cf. Bergson, Creative Evolution, p. 107: “Certain Foraminifera have not 
varied since the Silurian epoch. Unmoved witnesses of the innumerable revolu- 
tions that have upheaved our planet, the Lingulae are today what they were at 
the remotest times of the palaeozoic era.” 

+ Ray Lankester, A.M.N.H. (4), xt, p. 321, 1873. 


252 THE FORMS OF CELLS [cH. 


becomes much more manifest when we regard it from the point 
of view of physical and mathematical description and analysis, 
and whose form is referable, or (to. say the least of it) is very 
largely referable, to the direct and immediate action of a particular 
physical force. When we come to deal with the minute skeletons — 
of the Radiolaria we shall again find ourselves dealing with endless 
modifications of form, in which it becomes still more difficult to 
discern, or to apply, the guiding principle of affiliation or genealogy. 

Among the more aberrant forms of Infusoria is a little species 
known as Trichodina pediculus, a parasite on the Hydra, or fresh- 
water polype(Fig.81.) This Trichodina has the form of a more or less 
flattened circular disc, with a ring 
of cilia around both its upper and 
lower margins. The salient ridge 


a ey 
J 
NS NT from which these cilia spring may 


— 


\ Pe | be taken, as we have already said, 
JE aan to play the part of a strengthening 


po Cor ae “fillet.” The circular base of the 
WEES animal is flattened, in contact with 
the flattened surface of the Hydra 
over which it creeps, and the oppo- 
site, upper surface may be flattened nearly to a plane, or may at 
other times appear slightly convex or slightly concave. The sides 
of the little organism are contracted, forming a symmetrical 
equatorial groove between the upper and lower discs; and, on 
account of the minute size of the animal and its constant 
movements, we cannot submit the curvature of this concavity to 
measurement, nor recognise by the eye its exact contour. But 
it is evident that the conditions are precisely similar to those 
described on p. 223, where we were considering the conditions 
of stability of the catenoid. And it is further evident that, when 
the upper disc is actually plane, the equatorial groove is strictly 
a catenoid surface of revolution; and when on the other hand it 
is depressed, then the equatorial groove will tend to assume 
the form of a nodoidal surface. 
Another curious type is the flattened spiral of Dinenympha* 


Fig. 81. 


* Leidy, Parasites of the Termites, J. Nat. Sci., Philadelphia, vin, pp. 425— 
447, 1874-81; cf. Saville Kent’s Infusoria, u, p. 551. 


vy] OF COLLAR-CELLS 253 


which reminds us of the cylindrical spiral of a Spirillum among 
the bacteria. In Dinenympha we have a symmetrical figure, whose 
two opposite surfaces each constitute a surface of constant mean 
curvature; it is evidently a figure of equilibrium under certain 
special conditions of restraint. The cylindrical coil of the 
Spirillum, on the other hand, is a surface of constant mean curva- 
ture, and therefore of equilibrium, as truly, and in the same sense, 
as the cylinder itself. 


| IBR A 


\ 
Fig. 82. Dinenympha gracilis, Leidy. 


< 


A very curious conformation is that of the vibratile “collar,” 
found in Codosiga and the other ‘‘Choanoflagellates,’” and which 
we also meet with in the “collar-cells” which line the interior 
cavities of a sponge. Such collar-cells are always very minute, 
and the collar is constituted of a very delicate film, which 
shews an undulatory or rippling motion. It is a surface of 
revolution, and as it maintains itself in equilibrium (though a 
somewhat unstable and fluctuating one), it must be, under the 
restricted circumstances of its case, a surface of minimal area. 
But it is not so easy to see what these special circumstances are ; 
and it is obvious that the collar, if left to itself, must at once 


Te pice 


MEZiGos 


254. THE FORMS OF CELLS [cH. 


contract downwards towards its base, and become confluent with 
the general surface of the cell; for it has no 
longitudinal supports and no strengthening ring 
at its periphery. But in all these collar-cells, 
there stands within the annulus of the collar 
a large and powerful cilium or flagellum, in 
constant movement; and by the action of this 
flagellum, and doubtless in part also by the 
intrinsic vibrations of the collar itself, there is 
set up a constant steady current in the sur- 
rounding water, whose direction would seem to 
be such that it passes up the outside of the 
collar, down its inner side, and out in the middle 
in the direction of the flagellum; and there is a 
gy. 83. distinct eddy, in which foreign particles tend to 
be caught, around the peripheral margin of the collar. When the 
cell dies, that is to say when motion ceases, the collar immediately 
shrivels away and disappears. It is notable, by the way, that 
the edge of this little mobile cup is always smooth, never notched 
or lobed as in the cases we have discussed on p. 236: this latter 
condition being the outcome of a definite instability, marking the 
close of a period of equilibrium; while in the vibratile collar of 
Codosiga the equilibrium, such as it is, is being constantly: 
renewed and perpetuated like that of a juggler’s pole, by the 
motions of the system. I take it that, somehow, its existence 
(in a state of partial equilibrium) is due to the current motions, 
and to the traction exerted upon it through the friction of 
the stream which is constantly passing by. I think, in short, 
that it is formed very much in the same way as the cup-like ring 
of streaming ribbons, which we see fluttering and vibrating in the 
air-current of a ventilating fan. 

It is likely enough, however, that a different and much better 
explanation may yet be found: and if we turn once more to 
Mr Worthington’s Study of Splashes, we may find a curious 
suggestion of analogy in the beautiful craters encircling a central 
jet (as the collar of Codosiga encircles the flagellum), which we see 
produced in the later stages of the splash of a pebble*. 

* Op. cit. p. 79: 


iv] OF THE SIMPLER FORAMINIFERA 255 


Among the Foraminifera we have an immense variety of forms, 
which, in the light of surface tension and of the principle of 
minimal area, are capable of explanation and of reduction to a 
small number of characteristic types. Many of the Foraminifera 
are composite structures, formed by the successive imposition of 
cell upon cell, and these we shall deal with later on; let us glance 
here at the simpler conformations exhibited by the single cham- 
bered or “‘monothalamic”’ genera, and perhaps one or two of the 
simplest composites. 

We begin with forms, like Astrorhiza (Fig. 219, p. 464), which 
are in a high degree irregular, and end with others which manifest a 
perfect and mathematical regularity. The broad difference between 
these two types is that the former are characterised, like Amoeba, 
by a variable surface tension, and consequently by unstable equi- 
librium ; but the strong contrast between these and the regular forms 
is bridged over by various transition-stages, or differences of degree. 
Indeed, as in all other Rhizopods, the very fact of the emission of 
pseudopodia, which reach their highest development in this group 
of animals, is a sign of unstable surface-equilibrium ; and we must 
therefore consider that those forms which indicate symmetry and 
equilibrium in their shells have secreted these during periods when 
rest and uniformity of surface conditions alternated with the 
phases of pseudopodial activity. The irregular forms are in 
almost all cases arenaceous, that is to say they have no solid shells 
formed by steady adsorptive secretion, but only a looser covering 
of sand grains with which the protoplasmic body has come in 
contact and cohered. Sometimes, as in Ramulina, we have a 
calcareous shell combined with irregularity of form; but here we 
can easily see a partial and as it were a broken regularity, the 
recular forms of sphere and cylinder being repeated in various 
parts of the ramified mass. When we look more closely at the 
arenaceous forms, we find that the same thing is true of them; 
they represent, either in whole or part, approximations to the form 
of surfaces of equilibrium, spheres, cylinders and so forth. In 
Aschemonella we have a precise replica of the calcareous Ramulina ; 
and in Astrorhiza itself, in the forms distinguished by naturalists 
as A. crassatina, what is described as the “subsegmented interior *”’ 


* Brady, Challenger Monograph, pl. xx, p. 233. 


256 THE FORMS OF CELLS [cH.- 


seems to shew the natural, physical tendency of the long semifluid 
cylinder of protoplasm to contract, at its limit of stability, mto 
unduloid constrictions, as a step towards the breaking up into 
separate spheres: the completion of which process is restrained or 
prevented by the rigidity and friction of the arenaceous covering. 

Passing to the typical, calcareous-shelled Foraminifera, we have 
the most symmetrical of all possible types in the perfect sphere of 
Orbulina; this is a pelagic organism, whose floating habitat places. 


Fig. 84. Various species of Lagena. (After Brady.) 


it in a position of perfect symmetry towards all external forces. 
Save for one or two other forms which are also spherical, or 
approximately so, like Thurammina, the rest of the monothalamic 
caleareous Foraminifera are all comprised by naturalists within 
the genus Lagena. This large and varied genus consists of “flask- 
shaped” shells, whose surface is simply that of an unduloid, or 
more frequently, like that of a flask itself, an unduloid combined 
with a portion of a sphere. We do not know the circumstances 


Vv] OF HANGING DROPS 257 


under which the shell of Lagena is formed, nor the nature of the 
force by which, during its formation, the surface is stretched out 
into the unduloid form; but we may be pretty sure that it is 
suspended vertically in the sea, that is to say in a position of 
symmetry as regards its vertical axis, about which the unduloid 
surface of revolution is symmetrically formed. At the same time 
we have other types of the same shell in which the form is more 
or less flattened; and these are doubtless the cases in which such 
symmetry of position was not present, or was replaced by a broader, 
lateral contact with the surface pellicle*. 

While Orbulina is a simple spherical drop, Lagena suggests to 
our minds a “hanging drop,” drawn out to a long and slender 
neck by its own weight, aided by the viscosity of the material. 


(After Darling.) 


Indeed the various hanging drops, such as Mr C. R. Darling shews 
us, are the most beautiful and perfect unduloids, with spherical 
ends, that it is possible to conceive. A suitable liquid, a little 
denser than water and incapable of mixing with it (such as 
ethyl benzoate), is poured on a surface of water. It spreads 


* That the Foraminifera not only can but do hang from the surface of the 
water is confirmed by the following apt quotation which I owe to Mr EK. Heron- 
Allen: “Quand on place, comme il a été dit, le dép6t provenant du lavage des 
fucus dans un flacon que l’on remplit de nouvelle eau, on voit au bout d’une heure 
environ les animaux [Gromia dujardinii] se mettre en mouvement et commencer 
a grimper. Six heures apres ils tapissent l’extérieur du flacon, de sorte que les plus 
élevés sont a trente-six ou quarante-deux millimétres du fond; le lendemain 
beaucoup d’entre eux, aprés avoir atteint le niveau du liquide, ont continué a ramper 
@ sa surface, en se laissant pendre au-dessous comme certains mollusques gastéro- 
podes.” (Dujardin, F., Observations nouvelles sur les prétendus céphalopodes 
microscopiques, Ann. des Sci. Nat. (2), m1, p. 312, 1835.) 


lard 


TAGs 17 


258 THE FORMS OF CELLS [CH. 


over the surface and gradually forms a hanging drop, approxi- 
mately hemispherical; but as more liquid is added the drop 
sinks or rather grows downwards, still adhering to the surface 
film; and the balance of forces between gravity and surface 
tension results in the unduloid contour, as the increasing weight 
of the drop tends to stretch it out and finally break it in two. 
At the moment of rupture, by the way, a tiny droplet is formed 
in the attenuated neck, such as we described in the normal 
division of a cylindrical thread (p. 233). 

To pass to a much more highly organised class of animals, we find the 
unduloid beautifully exemplified in the little flask-shaped shells of certain 
Pteropod mollusca, e.g. Cuvierina*. Here again the symmetry of the figure 
would at once lead us to suspect that the creature lived in a position of 
symmetry to the surrounding forces, as for instance if it floated in the ocean 
in an erect position, that is to say with its long axis coincident with the direction 
of gravity; and this we know to be actually the mode of life of the little 
Pteropod. 

Many species of Lagena are complicated and beautified by a 
pattern, and some by the superaddition to the shell of plane 
extensions or “wings.” These latter give a secondary, bilateral 
symmetry to the little shell, and are strongly suggestive of a 
phase or period of growth in which it lay horizontally on the 
surface, instead of hanging vertically from the surface-film: in 
which, that is to say, it was a floating and not a hanging 
drop. The pattern is of two kinds. Sometimes it consists 
of a sort of fine reticulation, with rounded or more or 
less hexagonal interspaces: in other cases it is produced by a 
symmetrical series of ridges or folds, usually longitudinal, on the 
body of the flask-shaped cell, but occasionally transversely arranged 
upon the narrow neck. The reticulated and folded patterns we 
may consider separately. The netted pattern is very similar to the 
wrinkled surface of a dried pea, or to the more regular wrinkled 
patterns upon many other seeds and even pollen-grains. If a 
spherical body after developing a “skin” begin to shrink a little, 
and if the skin have so far lost its elasticity as to be unable to 
keep pace with the shrinkage of the inner mass, it will tend to 
fold or wrinkle; and if the shrinkage be uniform, and the elasticity 
and flexibility of the skin be also uniform, then the amount of 

* Cf. Boas, Spolia Atlantica, 1886, pl. 6. 


v] OF RETICULATE OR WRINKLED CELLS 259 


folding will be uniformly distributed over the surface. Little 
concave depressions will appear, regularly interspaced, and 
separated by convex folds. The little concavities being of equal 
size (unless the system be otherwise perturbed) each one will tend 
to be surrounded by six others; and when the process has reached 
its limit, the intermediate boundary-walls, or raised folds, will be 
found converted into a regular pattern of hexagons. 

But the analogy of the mechanical wrinkling of the coat of 
a seed is but a rough and distant one; for we are evidently dealing 
with molecular rather than with mechanical forces. In one of 
Darling’s experiments, a little heavy tar-oil is dropped onto a 
saucer of water, over which it spreads in a thin film showing 
beautiful interference colours after the fashion of those of a soap- 
bubble. Presently tiny holes appear in the film, which gradually 
increase in size till they form a cellular pattern or honeycomb, 
the oil gathering together in the meshes or walls of the cellular 
net. Some action of this sort is in all probability at work in a 
surface-film. of protoplasm covering the shell. As a physical 
phenomenon the actions involved are by no means fully under- 
stood, but surface-tension, diffusion and cohesion doubtless play 
their respective parts therein*. The very perfect cellular patterns 
obtained by Leduc (to which we shall have occasion to refer in 
a subsequent chapter) are diffusion patterns on a larger scale, but 
not essentially different. 

The folded or pleated pattern is doubtless to be explained, in 
a general way, by the shrinkage of a surface-film under certain 


r) 


* This cellular pattern would seem to be related to the ‘“‘cohesion figures’ 
described by Tomlinson in various surface-films (Phil. Mag. 1861 to 1870); to 
the “tesselated structure” in liquids described by Professor James Thomson in 
1882 (Collected Papers, p. 136); and to the tourbillons cellulaiies of Prof. H. Bénard 
(Ann. de Chimie (7), xxmm, pp. 62-144, 1901, (8), xxv, pp. 563-566, 1911), 
Rev. génér. des Sci. xt. p. 1268, 1900; cf. also E. H. Weber. Pcggend. Ann. 
xcrv, p. 452, 1855, etc.). The phenomenon is of great interest and various 
appearances have been referred to it, in biology, geology, metallurgy and even 
astronomy: for the flocculent clouds in the solar photosphere shew an analogous 
configuration. (See letters by Kerr Grant, Larmor, Wager and others, in Nature, 
April 16 to June 11, 1914.) In many instances, marked by strict symmetry or 
regularity, it is very possible that the interference of waves or ripples may play 
its part in the phenomenon. But in the majority of cases, it is fairly certain that 
localised centres of action, or of diminished tension, are present, such as might be 
provided by dust-particles in the case of Darling’s experiment (cf. infra, p. 590). 


17—2 


260 THE FORMS OF CELLS [on. 


conditions of viscous or frictional restraint. A case which (as it 
seems to me) is closely analogous to that of our foraminiferal 
shells is described by Quincke*, who let a film of albumin or of 
resin set and harden upon a surface of quicksilver, and found 
that the little solid pellicle had been 
thrown into a pattern of symmetrical 
folds. If the surface thus thrown into 
folds be that of a cylinder, or any other 
figure with one principal axis of sym- 
metry, such as an ellipsoid or unduloid, . 
the direction of the folds will tend to 
be related to the axis of symmetry, 
and we might expect accordingly to 
find regular longitudinal, or regular transverse wrinkling. Now 
as a matter of fact we almost invariably find in the Lagena 
the tormer condition: that is to say, in our ellipsoid or unduloid 
cell, the puckering takes the form of the vertical fluting on 
a column, rather than that of the transverse pleating of an 
accordion. And further, there is often a tendency for such 
longitudinal flutings to be more or less localised at the end of the 
ellipsoid, or in the region where the unduloid merges into its 
spherical base. In this latter region we often meet with a regular 
series of short longitudinal folds, as we do in the forms of Lagena 
denominated L. semistriata. All these various forms of surface 
can be imitated, or rather can be precisely reproduced, by the art 
of the glass-blower f. 

Furthermore, they remind one, in a striking way, of the 
regular ribs or flutings in the film or sheath which splashes up to 
envelop a smooth ball which has been dropped into a liquid, as 
Mr Wer hington has so beautifully shewn f. 


Fig 86. 


* Ueber physikalischen Eigenschaften diinner, fester Lamellen, S.B. Berlin. 
Akad. 1888, pp. 789, 790. 

+ Certain palaeontologists (e.g. Haeusler and Spandel) have maintained that 
in each family or genus the plain smooth-shelled forms are the primitive and ancient 
ones, and that the ribbed and otherwise ornamented shells make their appearance 
at later dates in the course of a definite evolution (cf. Rhumbler, Foraminiferen 
der Plavkton-Expedition, 1911, 1, p. 21). If this were true it would be of funda-— 
mental importance: but this book of mine would not deserve to be written. 

t A Study of Splashes, p. 116. 


v| OF FLUTED OR PLEATED CELLS 261 


In Mr Worthington’s experiment, there appears to be something 
of the nature of a viscous drag in the surface-pellicle; but whatever 
be the actual cause of variation of tension, it is not difficult to 
see that there must be in general a tendency towards longitudinal 
puckering or “‘fluting” in the case of a thin-walled cylindrical or 
other elongated body, rather than a tendency towards transverse 
puckering, or “pleating.” For let us suppose that some change 
takes place involving an increase of surface-tension in some small 
area of the curved wall, and leading therefore to an increase of 
pressure: that is to say let T become 7 + t, and P become P + p. 
Our new equation of equilibrium, then, in place of P= T/r + T/r’ 
becomes 


T P+t 
'B == P = — Be = ’ 
and by subtraction, 
p = t/r + tir’. 
Now if ree is ily > tir’. 


Therefore, in order to produce the small increment of pressure p, 
it is easier to do so by increasing ¢t/7 than t/r’; that is to say, the 
easier way is to alter, or diminish r. And the same will hold good 
if the tension and pressure be diminished instead of increased. 

This is as much as to say that, when corrugation or “rippling” 
of the walls takes place owing to small changes of surface-tension, 
and consequently of pressure, such corrugation is more likely to 
take place in the plane of 7,—that is to say, in the plane of greatest 
curvature. And it follows that in such a figure as an ellipsoid, 
wrinkling will be most likely to take place not only in a longitudinal 
direction but near the extremities of the figure, that is to say again 
in the region of greatest curvature. 

The longitudinal wrinkling of the flask-shaped bodies of our 
Lagenae, and of the more or less cylindrical cells of many other 
Foraminifera (Fig. 87), is in complete accord with the above theo- 
retical considerations; but nevertheless, we soon find that our result 
is not a general one, but is defined by certain limiting conditions, 
and is accordingly subject to what are, at first sight, important 
exceptions. For instance, when we turn to the narrow neck of 
the Lagena we see at once that our theory no longer holds; for 


262 THE FORMS OF CELLS [CH. 


the wrinkling which was invariably longitudinal in the body of 
the cell is as invariably transverse in the narrow neck. The reason 
for the difference is not far to seek. The conditions in the neck 
are very different from those in the expanded portion of the cell: 
the main difference being that the thickness of the wall is no longer 
insignificant, but is of considerable magnitude as compared with 
the diameter, or circumference, of the neck. We must accordingly 
take it into account in considering the bending moments at any 
point in this region of the shell-wall. And it is at once obvious 
that, in any portion of the narrow neck, flerwre of a wall in a 


Fig. 87. Nodosaria scalaris, Fig. 88. Gonangia of Campanularians. 
Batsch. (a) C. gracilis; (b) C. grandis. 
(After Allman.) 


transverse direction will be very difficult, while flexure in a 
longitudinal direction will be comparatively easy; just as, in the 
case of a long narrow strip of iron, we may easily bend it into 
folds running transversely to its long axis, but not the other way. 
The manner in which our little Lagena-shell tends to fold or wrinkle, 
longitudinally in its wider part, and transversely or annularly in 
its narrow neck, is thus completely and easily explained. 

An identical phenomenon is apt to occur in the little flask- 
shaped gonangia, or reproductive capsules, of some of the hydroid 
zoophytes. In the annexed drawings of these gonangia in two 
species of Campanularia, we see that in one case the little vesicle 


v] OF THE SIMPLER FORAMINIFERA 263 


has the flask-shaped or unduloid configuration of a Lagena; and 
here the walls of the flask are longitudinally fluted, just after the 
manner we have witnessed in the latter genus. But in the other 
Campanularian the vesicles are long, narrow and tubular, and here 
a transverse folding or pleating takes the place of the longitudinally 
fluted pattern. And the very form of the folds or pleats is 
enough to suggest that we are not dealing here with a simple 
phenomenon of surface-tension, but with a condition in which 
surface-tension and stiffness are both present, and play their 
parts in the resultant form. . 
Passing from the solitary flask-shaped cell of Lagena, we have, 
in another series of forms, a constricted cylinder, or succession 


a 


Fig. 89. Various Foraminifera (after Brady). a, Nodosaria simplex; 6, N. 
pygmaea; c, N. costulata; e, N. hispida; f, N. elata; d, Rheophax (Lituola) 
distans; g, Sagrina virgata. 


of unduloids; such as are represented in Fig. 89, illustrating 
certain species of Nodosaria, Rheophax and Sagrina. In some of 
these cases, and certainly in that of the arenaceous genus Rheophax, 
we have to do with the ordinary phenomenon of a segmenting or 
partially segmenting cylinder. But in others, the structure is 
not developed out of a continuous protoplasmic cylinder, but as 
we can see by examining the interior of the shell, it has been 
formed in successive stages, beginning with a simple unduloid 
“Lagena,” about whose neck, after its solidification, another drop 
of protoplasm accumulated, and in turn assumed the unduloid, 
or lagenoid, form. The chains of interconnected bubbles which 


264 THE FORMS OF CELLS [on. 


Morey and Draper made many years ago of melted resin are a 
very similar if not identical phenomenon*. 


There now remain for our consideration, among the Protozoa, 
the great oceanic group of the Radiolaria, and the little group of 
their freshwater allies, the Heliozoa. In nearly all these forms we 
have this specific chemical difference from the Foraminifera, that 
when they secrete, as they generally do secrete, a hard skeleton, 
it is composed of silica instead of lime. These organisms and the 
various beautiful and highly complicated skeletal fabrics which 
they develop give us many interesting illustrations of physical 
phenomena, among which the manifestations of surface-tension 
are very prominent. But the chief phenomena connected with 
their skeletons we shall deal with in another place, under the head 
of spicular concretions. 

In a simple and typical Heliozoan, such as the Sun-animaleule, 
Actinophrys sol, we have a “drop” of protoplasm, contracted by 
its surface tension into a spherical form. Within the heterogeneous 
protoplasmic mass are more fluid portions, and at the surface 
which separates these from the surrounding protoplasm a similar 
surface tension causes them also to assume the form of spherical 
“vacuoles,” which in reality are little clear drops within the big 
one; unless indeed they become numerous and closely packed, in 
which case, instead of isolated spheres or droplets they will 
constitute a “froth,” their mutual pressures and tensions giving 
rise to regular configurations such as we shall study in the next 
chapter. One or more of such clear spaces may be what is called 
a “contractile vacuole”: that is to say, a droplet whose surface 
tension is in unstable equilibrium and is apt to vanish altogether, 
so that the definite outline of the vacuole suddenly disappears f. 
Again, within the protoplasm are one or more nuclei, whose own 
surface tension (at the surface between the nucleus and the 
surrounding protoplasm), has drawn them in turn into the shape 


* See Silliman’s Journal, i, p. 179, 1820; and ef. Plateau, op. cit. 1, pp. 134, 
461. 


+ The presence or absence of the contractile vacuole or vacuoles is one of the 
chief distinctions, in systematic zoology, between the Heliozoa and the Radiolaria. 
As we have seen on p. 165 (footnote), it is probably no more than a physical con- 
sequence of the different conditions of existence in fresh water and in salt. 


Vv] OF THE SUN-ANIMALCULES 265 


of spheres. Outwards through the protoplasm, and stretching far 
beyond the spherical surface of the cell, there run stiff linear 
threads of modified or differentiated protoplasm, replaced or 
reinforced in some cases by delicate siliceous needles. In either 
case we know little or nothing about the forces which lead to their 
production, and we do not hide our ignorance when we ascribe 
their development to a “radial polarisation” of the cell. In the 
case of the protoplasmic filament, we may (if we seek for a 
hypothesis), suppose that it is somehow comparable to a viscid 
stream, or “liquid vein,” thrust or squirted out from the body of 
the cell. But when it is once formed, this long and comparatively 
rigid filament is separated by a distinct surface from the neigh- 
bouring protoplasm, that is to say from the more fluid surface- 
protoplasm of the cell; and the latter begins to creep up the 
filament, just as water would creep up the interior of a glass tube, 
or the sides of a glass rod immersed in the liquid. It is the simple 
case of a balance between three separate tensions: (1) that between 
the filament and the adjacent protoplasm, (2) that between the 
filament and the adjacent water, and (3) that between the water 
and the protoplasm. Calling these tensions respectively T;,, Ty», 
and T',,,, equilibrium will be attained when the angle of contact 
between the fluid protoplasm and the filament is such that 


DT ee as 


cos a = It is evident in this case that the angle is 


fp 
a very small one. The precise form of the curve is somewhat 
different from that which, under ordinary circumstances, is assumed 
by a liquid which creeps up a solid surface, as water in contact 
with air creeps up a surface of glass; the difference being due to 
the fact that here, owing to the density of the protoplasm being 
practically identical with that of the surrounding medium, the 
whole system is practically immune from gravity. Under normal 
circumstances the curve is part of the “elastic curve”’ by which 
that surface of revolution is generated which we have called, 
after Plateau, the nodoid; but in the present case it is apparently 
a catenary. Whatever curve it be, it obviously forms a surface 
of revolution around the filament. 

Since the attraction exercised by this surface tension is 
symmetrical around the filament, the latter will be pulled equally 


266 THE FORMS OF CELLS [on. 


in all directions; in other words it will tend to be set normally 
to the surface of the sphere, that is to say radiating directly 
outwards from the centre. If the distance between two adjacent 
filaments be considerable, the curve will simply meet the filament 
at the angle a already referred to; but if they be sufficiently near 
together, we shall have a continuous catenary curve forming a 
hanging loop between one filament and the other. And when this 
is so, and the radial filaments are more or less symmetrically 
interspaced, we may have a beautiful system of honeycomb-like 
depressions over the surface of the organism, each cell of the 
honeycomb having a strictly defined geometric configuration. 
In the simpler Radiolaria, the spherical form of the entire 
organism is equally well-marked; and here, as also in the more 
complicated Heliozoa (such as Actinosphaerium), the organism is 


Fig. 90. A, Trypanosoma tineae (after Minchin); B, Spirochaeta anodontae 
(after Fantham). 


differentiated into several distinct layers, each boundary surface 
tending to be spherical, and so constituting sphere within sphere. 
One of these layers at least is close packed with vacuoles, forming 
an “alveolar meshwork,” with the configurations of which we shall 
attempt in another chapter to correlate the characteristic structure 
of certain complex types of skeleton. 


An exceptional form of cell, but a beautiful manifestation of 
surface-tension (or so I take it to be), occurs in Trypanosomes, those 
tiny parasites of the ‘blood that are associated with sleeping- 
sickness and many other grave or dire maladies. These tiny 
organisms consist of elongated solitary cells down one side of which 
runs a very delicate frill, or “undulating membrane,” the free 
edge of which is seen to be slightly thickened, and the whole of 


v] OF UNDULATING MEMBRANES = 2OR 


which undergoes rhythmical and beautiful wavy movements. 
When certain Trypanosomes are artificially cultivated (for instance ° 
T. rotatorvum, from the blood of the frog), phases of growth are 
witnessed in which the organism has no undulating membrane, 
but possesses a long cilium or “flagellum,” springing from near 
the front end, and exceeding the whole body in length*. Again, 
in T. lewisw, when it reproduces by “multiple fission,” the 
products of this division are likewise devoid of an undulating 
membrane, but are provided with a long free flagellum}. It is 


Fig. 91. A, Trichomonas muris, Hartmann; B, Trichomastix serpentis, Dobell; 
C, Trichomonas. angusta, Alexeieff. (After Kofoid.) 


a plausible assumption to suppose that, as the flagellum waves 
about, it comes to lie near and parallel to the body of the cell, 
and that the frill or undulating membrane is formed by the clear, 
fluid protoplasm of the surface layer springing up in a film to run 
up and along the flagellum, just as a soap-film would be formed in 
similar circumstances. 

This mode of formation of the undulating membrane or frill 
appears to be confirmed by the appearances shewn in Fig. 91. 


* Cf. Doflein, Lehrbuch der Protozoenkunde, 1911, p. 422. 
+ Cf. Minchin, Introduction to the Study of the Protozoa, 1914 p. 293, Fig. 127. 


268 THE FORMS OF CELLS [CH. 


Here we have three little organisms closely allied to the ordinary 
Trypanosomes, of which one, Trichomastix (B), possesses four 
flagella, and the other two, Trichomonas, apparently three only: 
the two latter possess the frill, which is lacking in the first*. But 
it is impossible to doubt that when the frill is present (as in A and 
C), its outer edge is constituted by the apparently missing flagellum 
(a), which has become attached to the body of the creature at the 
point c, near its posterior end; and all along its course, the super- 
ficial protoplasm has been drawn out into a film, between the 
flagellum (a) and the adjacent surface or edge of the body (0). 
Moreover, this mode of formation has been actually witnessed 
and described, though in a somewhat exceptional case. The little 
flagellate monad Herpetomonas is normally destitute of an undulat- 
ing membrane, but possesses a single long terminal flagellum. 
According to Dr D. L. Mackinnon, the cytoplasm in a certain stage 
of growth becomes somewhat “‘sticky,’ a phrase which we may 
in all probability interpret to mean that its surface tension is 
being reduced. For this stickiness is 
shewn in two ways. In the first place, 
the long body, in the course of its 
various bending movements, is apt to 
adhere head to tail (so to speak), giving 
a rounded or sometimes annular form 
to the organism, such as has also been 
described in certain species or stages 
of Trypanosomes. But again, the 
long flagellum, if it get bent back- 
wards upon the body, tends to adhere 
to its surface. ‘‘ Where the flagellum 
was pretty long and active, its efforts 
to continue movement under these 
Pale dndanioe tou aneeed abnormal conditions resulted in the 
Trypanosome. (After D. L. gradual lifting up from the cytoplasm 
Mackinnon.) 
of the body of a sort of pseudo- 
undulating membrane (Fig. 92). The movements of this structure 
were so exactly those of a true undulating membrane that it was 


* Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates, etc. Pr. 
Amer. Acad. of Arts and Sci. ul, pp. 289-378, 1915. 


v| OF UNDULATING MEMBRANES 269 


difficult to believe one was not dealing with a small, blunt 
trypanosome*.” This in short is a precise description of the 
mode of development which, from theoretical considerations 
alone, we should conceive to be the natural if not the only 
possible way in which the undulating membrane could come into 
existence. 

There is a genus closely allied to Trypanosoma, viz. Trypano- 
plasma, which possesses one free flagellum, together with an 
undulating membrane; and it resembles the neighbouring genus 
Bodo, save that the latter has two flagella and no undulating 
membrane. In like manner, Trypanosoma so closely resembles 
Herpetomonas that, when individuals ascribed to the former genus 
exhibit a free flagellum only, they are said to be in the “Her- 
petomonas stage.” In short all through the order, we have pairs 
of genera, which are presumed to be separate and distinct, viz. 
Trypanosoma-Herpetomonas, Trypanoplasma-Bodo, Trichomastix- 
Trichomonas, in which one differs from the other mainly if not 

solely in the fact that a free flagellum in the one is replaced by an 
undulating membrane in the other. We can scarcely doubt that 
the two structures are essentially one and the same. 

‘ The undulating membrane of a Trypanosome, then, according 
to our interpretation of it, is a liquid film and must obey the law 
of constant mean curvature. It is under curious limitations of 
freedom: for by one border it is attached to the comparatively 
motionless body, while its free border is constituted by a flagellum 
which retains its activity and is bemg constantly thrown, like the 
lash of a whip, nto wavy curves. It follows that the membrane, 
for every alteration of its longitudinal curvature, must at the same 
instant become curved in a direction perpendicular thereto; it 
bends, not as a tape bends, but with the accompaniment of beautiful 
‘but tiny waves of double curvature, all tending towards the 
establishment of an “equipotential surface” ; and its characteristic 
undulations are not originated by an active mobility of the 
membrane but are due to the molecular tensions which produce 
the very same result in a soap-film under similar circumstances. 

In certain Spirochaetes, S. anodontae (Fig. 90) and S. balbiani 


_* D. L. Mackinnon, Herpetomonads from the Alimentary Tract of certain 
Dunegflies, Parasitology, 111, p. 268, 1910. 


270 THE FORMS OF CELLS [CH. 


(which we find in oysters), a very similar undulating membrane 
exists, but it is coiled in a regular spiral round the body of the cell. 
It forms a “screw-surface,” or helicoid, and, though we might 
think that nothing could well be more curved, yet its mathematical 
properties are such that it constitutes a ‘‘ruled surface” whose 
“mean curvature” is everywhere nil; and this property (as we 
have seen) it shares with the plane, and with the plane alone. 
Precisely such a surface, and of exquisite beauty, may be 
produced by bending a wire upon itself so that part forms an 
axial rod and part a spiral wrapping round the axis, and then 
dipping the whole into a soapy solution. 

These undulating and helicoid surfaces are exactly reproduced 
among certain forms of spermatozoa. The tail of a spermatozoon 
consists normally of an axis surrounded by clearer and more fluid 
protoplasm, and the axis sometimes splits up into two or more 
slender filaments. To surface tension operating between these 
and the surface of the fluid protoplasm (just as in the case of the 
flagellum of the Trypanosome), I ascribe the formation of the 
undulating membrane which we find, for instance, in the spermato- 
zoa of the newt or salamander; and of the helicoid membrane, 
wrapped in a far closer and more beautiful spiral than that which 
we saw in Spirochaeta, which is characteristic of the spermatozoa 
of many birds. 


Before we pass from the subject of the conformation of the 
solitary cell we must take some account of certain other exceptional 
forms, less easy of explanation, and still less perfectly understood. 
Such is the case, for instance, with the red blood-corpuscles of man 
and other vertebrates; and among the sperm-cells of the decapod 
crustacea we find forms still more aberrant and not less perplexing. 
These are among the comparatively few cells or cell-like structures. 
whose form seems to be incapable of explanation by theories of 
surface-tension. 

In all the mammalia (save a very few) the red blood-corpuscles 
are flattened circular discs, dimpled in upon their two opposite 
sides. This configuration closely resembles that of an india- 
rubber ball when we pinch it tightly between finger and thumb; 
and we may also compare it with that experiment of Plateau’s 


v] OF BLOOD CORPUSCLES 271 


(described on p. 223), where a flat cylindrical oil-drop, of certain 
relative dimensions, can, by sucking away a little of the contained 
oil, be made to assume the form of a biconcave disc, whose periphery 
is part of a nodoidal surface. From the relation of the nodoid 
to the “elastic curve,’ we perceive that these two examples are 
closely akin one to the other. 

The form of the corpuscle is symmetrical, and its surface is 
a surface of revolution; but it 
is obviously not a surface of Ek 1G 
constant mean curvature, nor of 
constant pressure. For we see 
at once that, in the sectional 
diagram (Fig. 93), the pressure 
inwards due to surface tension 
is positive at A, and negative at C; at B there is no 
curvature in the plane of the paper, while perpendicular to 
it the curvature is negative, and the pressure therefore is also 
negative. Accordingly, from the point of view of surface tension 
alone, the blood-corpuscle is not a surface of equilibrium; or in 
other words, it is not a fluid drop suspended in another liquid. 
It is obvious therefore that some other force or forces must be 
at work, and the simple effect of mechanical pressure is here 
excluded, because the blood-corpuscle exhibits its characteristic 
shape while floating freely in the blood. In the lower vertebrates 
the blood-corpuscles have the form of a flattened oval disc, with 
rather sharp edges and ellipsoidal surfaces, and this .again is 
manifestly not a surface of equilibrium. 

Two facts are especially noteworthy in connection with the 
form of the blood-corpuscle. In the first place, its form is only 
maintained, that is to say it is only in equilibrium, in relation to 
certain properties of the medium in which it floats. If we add a 
little water to the blood, the corpuscle quickly loses its character- 
istic shape and becomes a spherical drop, that is to gay a true 
surface of minimal area and of stable equilibrium. If on the other 
hand we add a strong solution of salt, or a little glycerine, the 
corpuscle contracts, and its surface becomes puckered and uneven. 
In these phenomena it is so far obeying the laws of diffusion and 
of surface tension. 


Fig. 93. 


272 THE FORMS OF CELLS [CH. 


In the second place, it can be exactly imitated artificially by 
means of other colloid substances. Many years ago Norris made the 
very interesting observation that in an emulsion of glue the drops 
assumed a biconcave form resembling that of the mammalian cor- 
puscles*. The glue was impure, and doubtless contained lecithin ; 
and it is possible (as Professor Waymouth Reid tells me) to make 
a similar emulsion with cerebrosides and cholesterin oleate, in 
which the same conformation of the drops or particles is beautifully 
shewn. Now such cholesterin bodies have an important place 
among those in which Lehmann and others have shewn and studied 
the formation of fluid crystals, that is to say of bodies in which 
the forces of crystallisation and the forces of surface tension are 
battling with one another}; and, for want of a better explanation, 
we may in the meanwhile suggest that some such cause is at the 
bottom of the conformation the explanation of which presents so 
many difficulties. But we must not, perhaps, pass from this 
subject without adding that the case is a difficult and complex 
one from the physiological point of view. For the surface of a 
blood-corpuscle consists of a “semi-permeable membrane,” through 
which certain substances pass freely and not others (for the most 
part anions and not cations), and it may be, accordingly, that we 
have in life a continual state of osmotic inequilibrium, of negative 
osmotic tension within, to which comparatively simple cause the 
imperfect distension of the corpuscle may be also duet. The whole 
phenomenon would be comparatively easy to understand if we 
might postulate a stiffer peripheral region to the corpuscle, in the 
form for instance of a peripheral elastic ring. Such an annular 
thickening or stiffening, like the “ collapse-rings”” which an engineer 
inserts in a boiler, has been actually asserted to exist, but its 
presence is not authenticated. . 

But it is not at all improbable that we have still much to 
learn about the phenomena of osmosis itself, as manifested in the 
case of minute bodies such as a blood-corpuscle; and (as Professor 
Peddie suggests to me) it is by no means impossible that curvature 


* Proc Roy. Soc. xu, pp. 251—257, 1862-3. 

+ Cf. (int. al.) Lehmann, Ueber scheinbar lebende Kristalle und Myelinformen, 
Arch. f. Entw. Mech. xxvi, p. 483, 1908; Ann. d. Physik, xiv, p 969, 1914. 

t Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the Red 
Blood Corpuscle, Biochem. Journal, 1, p. 55, 1908. 


vi OF CERTAIN OSMOTIC PHENOMENA 273 


of the surface may itself modify the osmotic or perhaps the adsorp- 
tive action. If it should be found that osmotic action tended to 
stop, or to reverse, on change of curvature, it would follow that 
this phenomenon would give rise to internal currents; and the 
change of pressure consequent on these would tend to intensify 
the change of curvature when once started*. — 

The sperm-cells of the Decapod crustacea exhibit various 
singular shapes. In the Crayfish they are flattened cells with 
‘stiff curved processes radiating outwards like a St Catherine’s 
wheel; in Inachus there are two such circles of stiff processes ; 
in Galathea we have a still more complex form, with long and 


a b c 


Fig. 94. Sperm-cells of Decapod Crustacea (after Koltzoff). a, Inachus scorpio; 
b, Galathea squamifera; c, do. after maceration, to shew spiral fibrillae. 


slightly twisted processes. In all these cases, just as in the case 
of the blood-corpuscle, the structure alters, and finally loses, its 
characteristic form when the nature or constitution (or as we may 
assume in particular—the density) of the surrounding medium is 
changed. 

Here again, as in the blood-corpuscle, we have to do with a 
very important force, which we had not hitherto considered in this 
connection,—the force of osmosis, manifested under conditions 
similar to those of Pfeffer’s classical experiments on the plant-cell. 
The surface of the cell acts as a “semi-permeable membrane,” 


* For an attempt to explain the form of a blood-corpuscle by surface-tension 
alone, see Rice, Phil. Mag. Nov. 1914; but cf. Shorter, ibid. Jan. 1915. 


Gis 18 


274. THE FORMS OF CELLS [cH. 


permitting the passage of certain dissolved substances (or their 
“ions”’) and including or excluding others; and thus rendering 
manifest and measurable the existence of a definite “osmotic 
pressure.” In the case of the sperm-cells of Inachus, certain 
quantitative experiments have been performed*. ‘The sperm-cell 
exhibits its characteristic conformation while lying in the serous 
fluid of the animal’s body, in ordinary sea-water, or in a 5 per 
cent. solution of potassium nitrate; these three fluids being all 
“isotonic” with one another. As we alter the concentration of 
potassium nitrate, the cell assumes certain definite forms corre- 
sponding to definite concentrations of the salt; and, as a further 
and final proof that the phenomenon is entirely physical, it is 
found that other salts produce an identical effect when their 
concentration is proportionate to their molecular weight, and 


KNOg 
© ©G BG 7 
We 125% “2% Pe a 
a b c d e 


Fig. 95. Sperm-cells of Inachus, as they appear in saline solutions of 
varying density. (After Koltzoff.) 


whatever identical effect is produced by various salts in their 
respective concentrations, a similarly identical effect is produced 
when these concentrations are doubled or otherwise proportionately 
changed f. 

Thus the following table shews the percentage concentrations 
of certain salts necessary to bring the cell into the forms a and ¢ 
of Fig. 95; in each case the quantities are proportional to the 
molecular weights, and in each case twice the quantity is necessary 
to produce the effect of Fig. 95¢ compared with that which gives 
rise to the all but spherical form of Fig. 95a. 


* Koltzoff, N. K., Studien tiber die Gestalt der Zelle, Arch. f. mikrosk. Anat. 
LXvu, pp. 364-571, 1905; Biol. Centralbl. xxi, pp. 680-696, 1903, xxv1, 
pp. 854-863, 1906; Arch. f. Zellforschung, u, pp. 1-65, 1908, vu, pp. 344-423, 
1911; Anat. Anzeiger, xu1, pp. 183-206, 1912. 

7 Cf. supra, p. 129. 


Vv] OF CERTAIN OSMOTIC PHENOMENA 275 


% concentration of salts in which 
the sperm-cell of Inachus 
assumes the form of 

———— 


fig. a fig. ¢ 
Sodium chloride 0:6 1-2 
Sodium nitrate 0-85 1:7 
Potassium nitrate 1-0 2-0 
Acétic acid 2-2 4-5 
Cane sugar 5-0 10-0 


If we look then, upon the spherical form of the cell as its true 
condition of symmetry and of equilibrium, we see that what we 
call its normal appearance is just one of many intermediate phases 
of shrinkage, brought about by the abstraction of fluid from its 
interior as the result of an osmotic pressure greater outside than 
inside the cell, and where the shrinkage of volume is not kept 
pace with by a contraction of the swrface-area. In the case of the 
blood-corpuscle, the shrinkage is of no great amount, and the 
resulting deformation is symmetrical; such structural inequality 
as may be necessary to account for it need be but ‘small. But 
in the case of the sperm-cells, we must have, and we actually do 
find, a somewhat complicated arrangement of more or less rigid 
or elastic structures in the wall of the cell, which like the wire 
framework in Plateau’s experiments, restrain and modify the 
forces acting on the drop. In one form of Plateau’s experiments, 
instead of supporting his drop on rings or frames of wire, he laid 
upon its surface one or more elastic coils; and then, on with- 
drawing oil from the centre of his globule, he saw its uniform 
shrinkage counteracted by the spiral springs, with the result that 
the centre of each elastic coil seemed to shoot out into a prominence. 
Just such spiral coils are figured 
(after Koltzoff) in Fig. 96; and they 
may be regarded as precisely akin to 
those local thickenings, spiral and 
other, to which we have already 
ascribed the cylindrical form of the 
Spirogyra cell. In all probability we 
must in like manner attribute the 
peculiar spiral and other forms, for 
instance of many Infusoria, to the 


Fig. 96. Sperm-cell of Dromia. 
(After Koltzoff. ) 


18—2 


276 THE FORMS OF CELLS [CH. V 


presence, among the multitudinous other differentiations of their 
protoplasmic substance, of such more or less elastic fibrillae, 
which play as it were the part of a microscopic skeleton*. 


But these cases which we have just dealt with, lead us to 
another consideration. In a semi-permeable membrane, through 
which water passes freely in and out, the conditions of a liquid 
surface are greatly modified; and, in the ideal or ultimate case, 
there is neither surface nor surface tension at all. And this would 
lead us somewhat to reconsider our position, and to enquire 
whether the true surface tension of a liquid film is actually 
responsible for all that we have ascribed to it, or whether certain 
of the phenomena which we have assigned to that cause may not 
in part be due to the contractility of definite and elastic membranes. 
But to investigate this question, in particular cases, is rather for 
the physiologist: and the morphologist may go on his way, 
paying little heed to what is no doubt a difficulty. In surface 
tension we have the production of a film with the properties of an 
elastic membrane, and with the special peculiarity that contraction 
continues with the same energy however far the process may have 
already gone; while the ordinary elastic membrane contracts to 
a certain extent, and contracts no more. But within wide limits 
the essential phenomena are the same in both cases. Our 
fundamental equations apply to both cases alike. And accord- 
ingly, so long as our purpose is morphological, so long as what we 
seek to explain is regularity and definiteness of form, it matters 
little if we should happen, here or there, to confuse surface tension 
with elasticity, the contractile forces manifested at a liquid 
surface with those which come into play at the complex internal 
surfaces of an elastic solid. 


* As Bethe points out (Zellgestalt, Plateausche Fliissigkeitsfigur und Neuro- 
fibrille, Anat. Anz. xu. p. 209, 1911), the spiral fibres of which Koltzoff speaks must 
lie in the surface, and not within the substance, of the cell whose conformation is 
affected by them, 


CHAPTER VI 


/ 


A NOTE ON ADSORPTION 


A very important corollary to, or amplification of the theory 
of surface tension is to be found in the modern chemico-physical 
doctrine of Adsorption*. In its full statement this subject soon 
becomes complicated, and involves physical conceptions and 
mathematical treatment which go beyond our range. But it is | 
necessary for us to take account of the phenomenon, though it 
be in the most elementary way. 

In the brief account of the theory of surface tension with which 
our last chapter began, it was pointed out that, in a drop of liquid, 
the potential energy of the system could be diminished, and work 
manifested accordingly, in two ways. In the first place we saw 
that, at our liquid surface, surface tension tends to set up an 
equilibrium of form, in which the surface is reduced or contracted 
either to the absolute minimum of a sphere, or at any rate to the 
least possible area which is permitted by the various circumstances 
and conditions; and if the two bodies which comprise our system, 
namely the drop of liquid and its surrounding medium, be simple 
substances, and the system be uncomplicated by other distributions 
of force, then the energy of the system will have done its work 
when this equilibrium of form, this minimal area of surface, is 
once attained. This phenomenon of the production of a minimal 
surface-area we have now seen to be of fundamental importance 
in the external morphology of the cell, and especially (so far 
as we have yet gone) of the solitary cell or unicellular organism. 

* See for a further but still elementary account, Michaelis, Dynamics Surfaces, 
1914, p. 22 seg.; Macallum, Oberfldchenspannung und Lebenserscheinungen, in 
Asher-Spiro’s Ergebnisse der Physiologie, x1, pp. 598-658, 1911; see also W. W. 
Taylor’s Chemistry of Colloids, 1915, p. 221 seq., Wolfgang Ostwald, Grundriss der 


Kolloidchemie, 1909, and other text-books of physical chemistry; and Bayliss’s 
Principles of General Physiology, pp. 54-73, 1915. 


278 . A NOTE ON ADSORPTION _ [CH- 


But we also saw, according to Gauss’s equation, that the 
potential energy of the system will be diminished (and its diminu- 
tion will accordingly be manifested in work) if from any cause 
the specific surface energy be diminished, that is to say if it be 
brought more nearly to an equality with the specific energy of the 
molecules in the interior of the lquid mass. This latter is a 
phenomenon of great moment in modern physiology, and, while 
we need not attempt to deal with it in detail, it has a bearing on 
cell-form and cell-structure which we cannot afford to overlook. 

In various ways a diminution of the surface energy may be © 
brought about. For instance, it is known that every isolated drop 
of fluid has, under normal circumstances, a surface-charge of 
electricity: in such a way that a positive or negative charge (as 
the case may be) is inherent in the surface of the drop, while a 
corresponding charge, of contrary sign, is inherent in the 
immediately adjacent molecular layer of the surrounding medium. 
Now the effect of this distribution, by which all the surface 
molecules of our drop are similarly charged, is that by virtue of 
this charge they tend to repel one another, and possibly also to 
draw other molecules, of opposite charge, from the interior of the 
mass; the result being in either case to antagonise or cancel, 
more or less, that normal tendency of the surface molecules to 
attract one another which is manifested in surface tension. In 
other words, an increased electrical charge concentrating at the 
surface of a drop tends, whether it be positive or negative, to 
lower the surface tension. 

But a still more important case has next to be considered. 
Let us suppose that our drop consists no longer of a single chemical 
substance, but contains other substances either in suspension or 
in solution. Suppose (as a very simple case) that it be a watery 
fluid, exposed to air, and containing droplets of oil: we know that 
the specific surface tension of oil in contact with air is much less 
than that of water, and it follows that, if the watery surface of 
our drop be replaced by an oily surface the specific surface energy 
of the system will be notably diminished. Now under these 
circumstances it is found that (quite apart from gravity, by which 
the oil might float to the surface) the oil has a tendency to be 
drawn to the surface; and this phenomenon of molecular attraction 


vi] ON SURFACE-CONCENTRATION 258 


or “adsorption” represents the work done, equivalent to the 
diminished potential energy of the system*. In more general 
terms, if a liquid (or one or other of two adjacent liquids) be a 
chemical mixture, some one constituent in which, if it entered 
into or increased in amount in the surface layer, would have the 
effect of diminishing its surface tension, then that constituent will 
have a tendency to accumulate or concentrate at the surface: the 
surface tension may be said, as it were, to exercise an attraction 
on this constituent substance, drawing it into the surface layer, 
and this tendency will proceed until at a certain “surface con- 
centration” equilibrium is reached, its opponent being that osmotic 
force which tends to keep the substance in uniform solution or 
diffusion. 

In the complex mixtures which constitute the protoplasm of 
the living cell, this phenomenon of “adsorption” has abundant 
play: for many of these constituents, such as oils, soaps, albumens, 
etc. possess the required property of diminishing surface tension. 

Moreover, the more a substance has the power of lowering the 
surface tension of the liquid in which it happens to be dissolved, 
the more will it tend to displace another and less effective substance 
from the surface layer. Thus we know that protoplasm. always 
contains fats or oils, not only in visible drops, but also in the 
finest suspension or “colloidal solution.” If under any impulse, 
such for instance as might arise from the Brownian movement, 
a droplet of oil be brought close to the surface, it is at once drawn 
into that surface, and tends to spread itself in a thin layer over 
the whole surface of the cell. But a soapy surface (for instance) 
would have in contact with the surrounding water a surface tension 
even less than that of the film of oil: and consequently, if soap 
be present in the water it will in turn be adsorbed, and will tend 
to displace the oil from the surface pellicle+. And this is all as 


* The first instance of what we now call an adsorptive phenomenon was 
observed in soap-bubbles. Leidenfrost, in 1756, was aware that the outer layer 
of the bubble was covered by an “oily” layer. A hundred years later Dupré 
shewed that in a soap-solution the soap tends to concentrate at the surface, so 
that the surface-tension of a very weak solution is very little different from that 
of a strong one (Théorie mécanique de la chaleur, 1869, p. 376; cf. Plateau, 1, 
p- 100). 

+ This identical phenomenon was the basis of Quincke’s theory of amoeboid 


280 A NOTE ON ADSORPTION [CH. 


much as to say that the molecules of the dissolved or suspended 
substance or substances will so distribute themselves throughout 
the drop as to lead towards an equilibrium, for each small unit 
of volume, between the superficial and internal energy; or so, in 
other words, as to lead towards a reduction to a minimum of the 
potential energy of the system. This tendency to concentration 
at the surface of any substance within the cell by which the surface 
tension tends to be diminished, or vice versa, constitutes, then, 
the phenomenon of Adsorption; and the general statement by 
which it is defined is known as the Willard-Gibbs, or Gibbs- 
Thomson law*. 

Among the many important physical features or concomitants 
of this phenomenon, let us take note at present that we need 
not conceive of a strictly superficial distribution of the adsorbed 
substance, that is to say of its direct association with the surface 
layer of molecules such as we imagined in the case of the electrical 
charge; but rather of a progressive tendency to concentrate, 
more and more, as the surface is nearly approached. Indeed we 
may conceive the colloid or gelatinous precipitate in which, in the 
case of our protoplasmic cell, the dissolved substance tends often 
to be thrown down, to constitute one boundary layer after another, 
the general effect being intensified and multiplied by the repeated 
addition of these new surfaces. 

Moreover, it is not less important to observe that the process 
of adsorption, in the neighbourhood of the surface of a hetero- 
geneous liquid mass, is a process which takes time; the tendency 
to surface concentration is a gradual and progressive one, and will 
fluctuate with every minute change in the composition of our 
substance and with every change in the area of its surface. In 
other words, it involves (in every heterogeneous substance) a 
continual instability of equilibrium: and a constant manifestation 


movement (Ueber periodische Ausbreitung von Fliissigkeitsoberflachen, etc., SB. 
Berlin. Akad. 1888, pp. 791-806; ef. Pfliiger’s Archiv, 1879, p. 136). 

* J. Willard Gibbs, Equilibrium of Heterogeneous Substances, 7'r. Conn. Acad. 
rit, pp. 380-400, 1876, also in Collected Papers, 1, pp. 185-218, London, 1906; 
J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888 (Surface 
tension of solutions), p. 190. See also (int. al.) the various papers by C. M. Lewis, 
Phil. Mag. (6), xv, p. 499, 1908, xvu1, p. 466, 1909, Zeitschr. f. physik. Chemie, 
LXx, p. 129, 1910; Milner, Phil. Mag. (6), x1, p. 96, 1907, etc. 


———— os 


vi] THE GIBBS-THOMSON LAW 281 


of motion, sometimes in the mere invisible transfer of molecules 
but often in the production of visible currents of fluid or manifest 
alterations in the form or outline of the system. 


_ The physiologist, as we have already remarked, takes account 
of the general phenomenon of adsorption in many ways: particu- 
larly in connection with various results and consequences of 
osmosis, inasmuch as this process is dependent on the presence 
of a membrane, or membranes, such as the phenomenon of adsorp- 
tion brings into existence. For instance it plays a leading part 
in all modern theories of muscular contraction, in which phenome- 
non a connection with surface tension was first indicated by 
FitzGerald and d’Arsonval nearly forty years ago*. And, as 
W. Ostwald was the first to shew, it gives us an entirely new 
conception of the relation of gases (that is to say, of oxygen and 
carbon dioxide) to the red corpuscles of the blood 7. 

But restricting ourselves, as much as may be, to our morpho- 
logical aspect of the case, there are several ways in which adsorption 
begins at once to throw light upon our subject. 

In the first place, our preliminary account, such as it is, 1s 
already tantamount to a description of the process of develop- 
ment of a cell-membrane, or cell-wall. The so-called “ secretion” 
of this cell-wall is nothing more than a sort of exudation, or 
striving towards the surface, of certain constituent molecules or 
particles within the cell; and the Gibbs-Thomson law formulates, 
in part at least, the conditions under which they do so. The 
adsorbed material may range from the almost unrecognisable 
pellicle of a blood-corpuscle to the distinctly differentiated 
“ectosarc’”’ of a protozoan, and again to the development of a 
fully formed cell-wall, as in the cellulose partitions of a vegetable 
tissue. In such cases, the dissolved and adsorbable material has 
not only the property of lowering the surface tension, and hence 


* G. F. FitzGerald, On the Theory of Muscular Contraction, ie Ass. Rep. 
1878; also in Scientific Writings, ed. Larmor, 1902, pp. 34. 75. A. d’Arsonval, 
Relations entre l’électricité animale et la tension superficielle, C. R. cv1, p. 1740, 
1888; cf. A Imbert, Le mécanisme de la contraction musculaire, déduit de la con- 
sidération des forces de tension superficielle, Arch. de Phys. (5), 1X. pp. 289-301, 1897. 

+ Ueber die Natur der Bindung der Gase im Blut und in seinen Bestandtheilen, 
Kolloid. Zeitschr. 1, pp. 264-272, 294-301, 1908; cf. Loewy, Dissociationsspan- 
nung des Oxyhaemoglobin im Blut, Arch. f. Anat. und Physiol. 1904, p. 231. 


\ 
282 A NOTE ON ADSORPTION [CH. 


of itself accumulating at the surface, but has also the property 
of increasing the viscosity and mechanical rigidity of the material 
in which it is dissolved or suspended, and so of constituting 
a visible and tangible “membrane*.” The “zoogloea” around a 
group of bacteria is probably a phenomenon of the same order. 
In the superficial deposition of inorganic materials we see the 
same process abundantly exemplified. Not only do we have the 
simple case of the building of a shell or “test” upon the outward 
surface of a living cell, as for instance in a Foraminifer, but in a 
subsequent chapter, when we come to deal with various spicules 
and spicular skeletons such as those of the sponges and of the 
Radiolaria, we shall see that it is highly characteristic of the 
whole process of spicule-formation for the deposits to be laid 
down just in the “interfacial” boundaries between cells or 
vacuoles, and that the form of the spicular structures tends in 
many cases to be regulated and determined by the arrangement 
of these boundaries. 


In physical chemistry, an important distinction is drawn between adsorption 
and pseudo-adsorption+, the former being a reversible, the latter an irreversible 
or permanent phenomenon. That is to say, adsorption, strictly speaking, 
implies the surface-concentration of a dissolved substance, under circumstances 
which, if they be altered or reversed, will cause the concentration to diminish 
or disappear. But pseudo-adsorption includes cases, doubtless originating in 
adsorption proper, where subsequent changes leave the concentrated substance 
incapable of re-entering the liquid system. It is obvious that many (though 
not all) of our biological illustrations, for instance the formation of spicules 
or of permanent cell-membranes, belong to the class of so-called pseudo- 
adsorption phenomena. But the apparent contrast between the two is in 
the main a secondary one, and however important to the chemist is of little 
consequence to us. 


* We may trace the first steps in the study of this phenomenon to Melsens, 
who found that thin films of white of egg become firm and insoluble (Sur les modi- 
fications apportées & l’albumine...par l’action purement mécanique, C. R. Acad. 
Sci. Xxxiu, p. 247, 1851); and Harting made similar observations about the same 
time. Ramsden has investigated the same subject, and also the more general 
phenomenon of the formation of albuminoid and fatty membranes by adsorption : 
cf. Koagulierung der Eiweisskérper auf mechanischer Wege, Arch. f. Anat. u. Phys. 
(Phys. Abth.) 1894, p. 517; Abscheidung fester Kérper in Oberfliichenschichten 
Z. f. phys. Chem. xiv, p. 341, 1902; Proc. R. S. uxxit, p. 156, 1904. For a general 
review of the whole subject see H. Zangger, Ueber Membranen und Membranfunk- 
tionen, in Asher-Spiro’s Ergebnisse der Physiologie, vu, pp. 99-160, 1908. 

+ Cf. Taylor, Chemistry of Colloids, p. 252. 


v1] THE FORMATION OF MEMBRANES 283 


While this brief sketch of the theory of membrane-formation 
is cursory and inadequate, it is enough to shew that the physical 
theory of adsorption tends in part to overturn, in part to simplify 
enormously, the older histological descriptions. We can no longer 
be content with such statements as that of Strasbiirger, that 
membrane-formation in general is associated with the “activity 
of the kinoplasm,” or that of Harper that a certain spore-membrane 
arises directly from the astral rays*. In short, we have easily 
reached the general conclusion that the formation of a cell-wall 
or cell-membrane is a chemico-physical phenomenon, which the 
purely objective methods of the biological microscopist do not 
suffice to interpret. 


If the process of adsorption, on which the formation of a 
membrane depends, be itself dependent on the power of the 
adsorbed substance to lower the surface tension, it is obvious that 
adsorption can only take place when the surface tension already 
present is greater than zero. It is for this reason that films or 
threads of creeping protoplasm shew little tendency, or none, to 
cover themselves with an encysting membrane; and that it is 
only when, in an altered phase, the protoplasm has developed 
a positive surface tension, and has accordingly gathered itself up 
into a more or less spherical body, that the tendency to form a 
membrane is manifested, and the organism develops its “cyst” 
or cell-wall. 

It is found that a rise of temperature greatly reduces the 
adsorbability of a substance, and this doubtless comes, either in 
part or whole, from the fact that a rise of temperature is itself 
a cause of the lowering of surface tension. We may in all pro- 
bability ascribe to this fact and to its converse, or at least associate 
with it, such phenomena as the encystment of unicellular organisms 
at the approach of winter, or the frequent formation of strong 
shells or membranous capsules in “ winter-eggs.”’ 

Again, since a film or a froth (which is a system of films) can 
only be maintained by virtue of a certain viscosity or rigidity of 


* Strasbtirger, Ueber Cytoplasmastrukturen, etc. Jahrb. f. wiss. Bot. xxx, 
1897; R. A. Harper, Kerntheilung und freie Zellbildung im Ascus, ibid.: cf. 
Wilson, The Cell in Development, etc. pp. 53-55. 


284 A NOTE ON ADSORPTION [CH. 


the liquid, it may be quickly caused to disappear by the presence 
in its neighbourhood of some substance capable of reducing the 
surface tension; for this substance, being adsorbed, may displace 
from the adsorptive layer a material to which was due the rigidity 
of the film. In this way a “bathytonic” substance such as ether 
causes most foams to subside, and the pouring oil on troubled 
waters not only stills the waves but still more quickly dissipates 
the foam of the breakers. The process of breaking up an alveolar 
network, such as occurs at a certain stage in the nuclear division 
of the cell, may perhaps be ascribed in part to such a cause, as 
well as to the direct lowering of surface tension by electrical 
agency. 

Our last illustration has led us back to the subject of a previous 
chapter, namely to the visible configuration of the interior of the 
cell; and in connection with this wide subject there are many 
phenomena on which light is apparently thrown by our knowledge 
of adsorption, and of which we took little or no account in our 
former discussion. One of these phenomena is that visible or 
concrete “ polarity,” which we have already seen to be in some way 
associated with a dynamical polarity of the cell. 

This morphological polarity may be of a very simple kind, as 
when, in an epithelial cell, it is manifested by the outward shape 
of the elongated or columnar cell itself, by the essential difference 
between its free surface and its attached base, or by the presence 
in the neighbourhood of the former of mucous or other products 
of the cell’s activity. But in a great many cases, this “ polarised” 
symmetry is supplemented by the presence of various fibrillae, or 
of linear arrangements of particles, which in the elongated or 
“monopolar” cell run parallel with its axis, and which tend to 
a radial arrangement in the more or less rounded or spherical 
cell. Of late years especially, an immense importance has been 
attached to these various linear or fibrillar arrangements, as they 
occur (after staining) in the cell-substance of intestinal epithelium, 
of spermatocytes, of ganglion cells, and most abundantly and 
most frequently of all in gland cells. Various functions, which 
seem somewhat arbitrarily chosen, have been assigned, and many 
hard names given to them; for these structures now include your 
mitochondria and your chondriokonts (both of these being varieties 


a 


vr] OF MORPHOLOGICAL POLARITY 285 


of chondriosomes), your Altmann’s granules, your microsomes, 
pseudo-chromosomes, epidermal fibrils and basal filaments, your 
archeoplasm and ergastoplasm, and probably your idiozomes, 
plasmosomes, and many other histological minutiae*. 

The position of these bodies with regard to the other cell- 
structures is carefully described. Sometimes they lie in the 
neighbourhood of the nucleus itself, that is to say in proximity to 
the fluid boundary surface which separates the nucleus from the 
cytoplasm; and in this position they often form a somewhat cloudy 
sphere which constitutes the Nebenkern. In the majority of cases, 
as in the epithelial cells, they form filamentous structures, and rows 
of granules, whose main direction is parallel to the axis of the 


7 
' 


1, 


A 


Fig. 97. A, B, Chondriosomes in kidney-cells, prior to and during secretory 
activity (after Barratt); C, do. in pancreas of frog (after Mathews). 


Un) 
B 


cell, and which may, in some cases, and in some forms, be con- 
spicuous at the one end, and in some cases at the other end of 
the cell. But I do not find that the histologists attempt to explain, 
or to correlate with other phenomena, the tendency of these bodies 
to lie parallel with the axis, and perpendicular to the extremities 
of the cell; itis merely noted as a peculiarity, or a specific character, 
of these particular structures. Extraordinarily complicated and 
diverse functions have been ascribed to them. Engelmann’s 
“Fibrillenkonus,” which was almost certainly another aspect of 
the same phenomenon, was held by him and by cytologists like 
Breda and Heidenhain, to be an apparatus connected in some 


* Cf. A. Gurwitsch, Morphologie und Biologie der Zelle, 1904, pp. 169-185; 
Meves, Die Chondriosomen als Traiger erblicher Anlagen, Arch. f. mikrosk. Anat. 
1908, p. 72; J. O. W. Barratt, Changes in Chondriosomes, etc. Q.J.M.S. Lv1i1, 
pp. 553-566, 1913, etc.; A. Mathews, Changes in Structure of the Pancreas 
Cell, etc., J. of Morph. xv (Suppl.), pp. 171-222, 1899. 


236 A NOTE ON ADSORPTION _ (on. 


unexplained way with the mechanism of ciliary movement. 
Meves looked upon the chondriosomes as the actual carriers or 
transmitters of heredity*. Altmann invented a new aphorism, 
Omne granulum e granulo, as a refinement of Virchow’s omnis 
cellula e cellula; and many other histologists, more or less in accord, 
accepted the chondriosomes as important entities, sui generis, 
intermediate in grade between the cell itself and its ultimate 
molecular components. The extreme cytologists of the Munich 
school, Popofi, Goldschmidt and others, following Richard Hertwig, 
declaring these structures to be identical with “chromidia” (under 
which name Hertwig ranked all extra-nuclear chromatin), would 
assign them complex functions in maintaining the balance between. 
nuclear and cytoplasmic material; and the “chromidial hypo- 
thesis,” as every reader of recent cytological hterature knows, has 
become a very abstruse and complicated thing}. With the help 
of the “binuclearity hypothesis” of Schaudinn and his school, it 
has given us the chromidial net, the chromidial apparatus, the 
trophochromidia, idiochromidia, gametochromidia, the protogono- 
plasm, and many other novel and original conceptions. The 
names are apt to vary somewhat in significance from one writer 
to another. 

The outstanding fact, as it seems to me, is that physiological 
science has been heavily burdened in this matter, with a_jargon 
of names and a thick cloud of hypotheses; while, from the physical 
point of view we are tempted to see but little mystery in the 
whole phenomenon, and to ascribe it, in all probability and in 
general terms, to the gathering or “clumping” together, under 
surface tension, of various constituents of the heterogeneous cell- 
content, and to the drawing out of these little clumps along the 
axis of the cell towards one or other of its extremities, in relation 
to osmotic currents, as these in turn are set up in direct relation 


* The question whether chromosomes, chondriosomes or chromidia be the true 
vehicles or transmitters of “heredity” is not without its analogy to the older problem 
of whether the pineal gland or the pituitary body were the actual seat and domicile 
of the soul. 

+ Cf. C. C. Dobell, Chromidia and the Binuclearity Hypotheses; a review and 
a criticism, Q.J.M.S. tot, 279-326, 1909; Prenant, A., Les Mitochondries et 
lErgastoplasme, Journ. de ? Anat. et de la Physiol. xuvi1, pp. 217-285, 1910 (both 
with copious bibliography). 


vr] MACALLUM’S EXPERIMENTS 287 


to the phenomena of surface energy and of adsorption*. And 
all this implies that the study of these minute structures, if it 
teach us nothing else, at least surely and certainly reveals to us 
the presence of a definite “field of force,” and a dynamical polarity 
within the cell. 


Our next and last illustration of the effects of adsorption, 
which we owe to the investigations of Professor Macallum, is of 
great importance; for it introduces us to a series of phenomena 
in regard to which we seem now to stand on firmer ground than 
in some of the foregoing cases, though we cannot yet consider that 
the whole story has been told. In our last chapter we were 
restricted mainly, though not entirely, to a consideration of figures 
of equilibrium, such as the sphere, the cylinder or the undulojd; 
and we began at once to find ourselves in difficulties when we were 
confronted by departures from symmetry, as for instance in the 
simple case of the ellipsoidal yeast-cell and the production of its 
bud. We found the cylindrical cell of Spirogyra, with its plane 
or spherical ends, a comparatively simple matter to understand ; 
but when this uniform cylinder puts out a lateral outgrowth, in 
the act of conjugation, we have a new and very different system 
of forces to explain. The analogy of the soap-bubble, or of the 
simple liquid drop, was apt to lead us to suppose that the surface 
tension was, on the whole, uniform over the surface of our cell: 
and that its departures from symmetry of form were therefore 
likely to be due to variations in external resistance. But if we 
have been inclined to make such an assumption we must now 


* Traube in particular has maintained that in differences of surface-tension 
we have the origin of the active force productive of osmotic currents, and that 
herein we find an explanation, or an approach to an explanation, of many phenomena 
which were formerly deemed peculiarly “vital” in their character. “Die Differenz 
der Oberflachenspannungen oder der Oberfliichendruck eine Kraft darstellt, welche 
als treibende Kraft der Osmose, an die Stelle des nicht mit dem Oberflichendruck 
identischen osmotischen Druckes, zu setzen ist, etc.” (Oberflachendruck und 
seine Bedeutung im Organismus, Pfliiger’s Archiv, ov, p. 559, 1904.) Cf. also 
Hardy (Pr. Phys. Soc. xxvut, p. 116, 1916), “If the surface film of a colloid 
membrane separating two masses of fluid were to change in such a way as to lower 
the potential of the water in it, water would enter the region from both sides at 
once. But if the change of state were to be propagated as a wave of change, 
starting at one face and dying out at the other face, water would be carried along 
from one side of the membrane to the other. A succession of such waves would 
maintain a flow of fluid.” 


288 A NOTE ON ADSORPTION [CH. 


reconsider it, and be prepared to deal with important localised 
variations in the surface tension of the cell. For, as a matter of 
fact, the simple case of a perfectly symmetrical drop, with uniform 
surface, at which adsorption takes place with similar uniformity, 
is probably rare in physics, and rarer still (if it exist at all) in the 
fluid or fluid-containing system which we call in biology a cell. 
We have mostly to do with cells whose general heterogeneity of 
substance leads to qualitative differences of surface, and hence to 
varying distributions of surface tension. We must accordingly 
investigate the case of a cell which displays some definite and 
regular heterogeneity of its liquid surface, just as Amoeba displays 
a heterogeneity which is complex, irregular and continually 
fluctuating in amount and distribution. Such heterogeneity as 
we are speaking of must be essentially chemical, and the prelimin- 
ary problem is to devise methods of “microchemical” analysis, 
which shall reveal localised accumulations of particular substances 
within the narrow limits of a cell, in the hope that, their normal 
effect on surface tension being ascertained, we may then correlate 
with their presence and distribution the actual indications of 
varying surface tension which the form or movement of the cell 
displays. In theory the method is all that we could wish, but in 
practice we must be content with a very limited application of it; 
for the substances which may have such action as we are looking 
for, and which are also actual or possible constituents of the cell, 
are very numerous, while the means are very seldom at hand to 
demonstrate their precise distribution and localisation. But in 
one or two cases we have such means, and the most notable is in 
connection with the element potassium. As Professor Macallum 
has shewn, this element can be revealed, in very minute quantities, 
by means of a certain salt, a mitrite of cobalt and sodium*. This 
salt penetrates readily into the tissues and into the interior of the 
cell; it combines with potassium to form a sparingly soluble 
nitrite of cobalt, sodium and potassium; and this, on subsequent 
treatment with ammonium sulphide, is converted into a character- 
istic black precipitate of cobaltic sulphide fy. 


* On the Distribution of Potassium in animal and vegetable Cells; Journ. of 


Physiol. Xxxi1, p. 95, 1905. 
+ The reader will recognise that there is a fundamental difference, and contrast, 


v1] MACALLUM’S EXPERIMENTS 289 


By this means Macallum demonstrated some years ago the 
unexpected presence of accumulations of potassium (i.e. of chloride 
or other salts of potassium) localised in particular parts of various 
cells, both solitary cells and tissue cells; and he arrived at the 
conclusion that the localised accumulations in question were 
simply evidences of concentration of the dissolved potassium salts, 
formed and localised in accordance with the Gibbs-Thomson law. 
In other words, these accumulations, occurring as they actually do 
in connection with various boundary surfaces, are evidence, when 
they appear irregularly distributed over such a surface, of in- 
equalities in its surface tension*; and we may safely take it that 
our potassium salts, like inorganic substances in general, tend to 
raise the surface tension, and will therefore be found concentrating 
at a portion of the surface whose tension is weak ft. 

In Professor Macallum’s figure (Fig. 98, 1) of the little green 
alga Pleurocarpus, we see that one side of the cell is beginning to 
bulge out in a wide convexity. This bulge is, in the first place, 
a sign of weakened surface tension on one side of the cell, which as 
a whole had hitherto been a symmetrical cylinder; in the second 
place, we see that the bulging area corresponds to the position of 
a great concentration of the potassic salt; while in the third place, 
from the physiological point of view, we call the phenomenon 
the first stage in the process of conjugation. In Fig. 98, 2, of 
Mesocarpus (a close ally of Spirogyra), we see the same phenomenon 
admirably exemplified in a later stage. From the adjacent cells 
distinct outgrowths are being emitted, where the surface tension has 
been weakened: just as the glass-blower warms and softens a small 
part of his tube to blow out the softened area into a bubble or 
diverticulum; and in our Mesocarpus cells (besides a certain 
amount of potassium. rendered visible over the boundary which 
between such experiments as these of Professor Macallum’s and the ordinary 
staining processes of the histologist. The latter are (as a general rule) purely- 
empirical, while the former endeavour to reveal the true microchemistry of the 
cell. “On peut dire que la microchimie n’est encore qu’& la période d’essai, et 
-que l'avenir de Vhistologie et spécialement de la cytologie est tout entier dans la 
microchimie”’ (Prenant, A., Méthodes et résultats de la Microchimie, Journ. de 
PAnat. et de la Physiol. xtvt, pp. 343-404, 1910). 


* Cf. Macallum, Presidential Address, Section I, Brit. Ass. Rep. (Sheffield), 
1910, p. 744. 


t In accordance with a simple corollary to the Gibbs-Thomson law. 


mG. 19 


290 A NOTE OF ADSORPTION [CH. 


separates the green protoplasm from the cell-sap), there is a very 
large accumulation precisely at the poimt where the tension of the 
originally cylindrical cell is weakening to produce the bulge. 
But in a still later stage, when the boundary between the two 
conjugating cells is lost and the cytoplasm of the two cells becomes 
fused together, then the signs of potassic concentration quickly 
disappear, the salt becoming generally diffused through the now 
symmetrical and spherical “zygospore.” 


Fig. 98. Adsorptive concentration of potassium salts in (1) cell of Plewrocarpus 
about to conjugate; (2) conjugating cells of Mesocarpus; (3) sprouting spores 
of Equisetum. (After Macallum.) 


In aspore of Equisetum (Fig. 98, 3), while it is still a single cell, 
no localised concentration of potassium is to be discerned; but as 
‘soon as the spore has divided, by an internal partition, into two 
cells, the potassium salt is found to be concentrated in the smaller 
one, and especially towards its outer wall, which is marked by a 
pronounced convexity. And as this convexity (which corresponds 
to one pole of the now asymmetrical, or quasi-ellipsoidal spore) 
grows out into the root-hair, the potassium salt accompanies its 
growth, and is concentrated under its wall. The concentration is, 


v1] MACALLUMW’S EXPERIMENTS 291 


accordingly, a concomitant of the diminished surface tension which 
is manifested in the altered configuration of the system. 

In the case of ciliate or flagellate cells, there is to be found a 
characteristic accumulation of potassium at and near the base o1 
the cilia. The relation of ciliary movement to surface tension 
lies beyond our range, but the fact which we have just mentioned 
throws light upon the frequent or general presence of a little 
protuberance of the cell-surface just where a flagellum is given 
off (cf. p. 247), and of a little projecting ridge or fillet at the base 
of an isolated row of cilia, such as we find in Vorticella. 

Yet another of Professor Macallum’s demonstrations, though 
its interest is mainly physiological, will help us somewhat further 
to comprehend what is implied in our phenomenon. In a normal 
cell of Spirogyra, a concentration of potassium is revealed along 
the whole surface of the spiral coil of chlorophyll-bearing, or 
“chromatophoral,” protoplasm, the rest of the cell being wholly 
destitute of the former substance: the indication being that, at 
this particular boundary, between chromatophore and cell-sap, 
the surface tension is small in comparison with any other interfacial 
surface within the system. 

Now as Macallum points out, the presence of potassium is 
known to be a factor, in connection with the chlorophyll-bearing 
protoplasm, in the synthetic production of starch from CO, under 
the influence of sunlight. But we are left in some doubt as to 
the consecutive order of the phenomena. For the lowered surface 
tension, indicated by the presence of the potassium, may be 
itself a cause of the carbohydrate synthesis; while on the other 
hand, this synthesis may be attended by the production of sub- 
stances (e.g. formaldehyde) which lower the surface tension, and 
so conduce to the concentration of potassium. All we know for 
certain is that the several phenomena are associated with one 
another, as apparently inseparable parts or inevitable concomitants 
of a certain complex action. 


And now to return, for a moment, to the question of cell-form. 
When we assert that the form of a cell (in the absence of mechanical 
pressure) is essentially dependent on surface tension, and even when 
we make the preliminary assumption that protoplasm is essentially 

192 


292 A NOTE OF ADSORPTION [CH. V1 


a fluid, we are resting our belief on a general consensus of evidence, 
rather than on compliance with any one crucial definition. The 
simple fact is that the agreement of cell-forms with the forms 
which physical experiment and mathematical theory assign to 
liquids under the influence of surface tension, is so frequently and 
often so typically manifested, that we are led, or driven, to accept 
the surface tension hypothesis as generally applicable and as 
equivalent to a universal law. The occasional difficulties or 
apparent exceptions are such as call for further enquiry, but fall 
short of throwimg doubt upon our hypothesis. Macallum’s 
researches introduce a new element of certainty, a “nail in a sure 
place,” when they demonstrate that, in certain movements or 
changes of form which we should naturally attribute to weakened 
surface tension, a chemical concentration which would naturally 
accompany such weakening actually takes place. They further 
teach us that in the cell a chemical heterogeneity may exist of 
a very marked kind, certain substances being accumulated here 
and absent there, within the narrow bounds of the system. 

Such localised accumulations can as yet only be demonstrated 
in the case of a very few substances, and of a single one in par- 
ticular; and these are substances whose presence does not produce, 
but whose concentration tends to follow, a weakening of surface 
tension. The physical cause of the localised inequalities of surface 
tension remains unknown. We may assume, if we please, that it 
is due to the prior accumulation, or local production, of chemical 
bodies which would have this direct effect; though we are by 
no means limited to this hypothesis. 

But in spite of some remaining difficulties and uncertainties, 
we have arrived at the conclusion, as regards unicellular organisms, 
that not only their general configuration but also their departures 
from symmetry may be correlated with the molecular forces 
manifested in their fluid or semi-fluid surfaces. 


CHAPTER VII 
THE FORMS OF TISSUES OR CELL-AGGREGATES 


We now pass from the consideration of the solitary cell to that 
of cells in contact with one another,—to what we may call in 
the first instance “cell-aggregates,’—through which we shall be led 
ultimately to the study of complex tissues. In this part of our 
subject, as in the preceding chapters, we shall have to give some 
consideration to the effects of various forces; but, as in the case 
- of the conformation of the solitary cell, we shall probably find, 
and we may at least begin by assuming, that the agency of surface 
tension is especially manifest and important. The effect of this 
surface tension will chiefly manifest itself in the production of 
surfaces minimae areae: where, as Plateau was always careful to 
point out, we must understand by this expression not an absolute, 
but a relative minimum, an area, that is to say, which approxi- 
mates to an absolute minimum as nearly as circumstances and the 
conditions of the case permit. 

There are certain fundamental principles, or fundamental 
equations, besides those which we have already considered, which 
we shall need in our enquiry. For instance the case which we 
briefly touched upon (on p. 265) of the angle of contact between 
the protoplasm and the axial filament in a Heliozoan we shall 
now find to be but a particular case of a general and elementary 
theorem. 

Let us re-state as follows, in terms of Energy, the general 
principle which underlies the theory of surface tension or capillarity. 

When a fluid is m contact with another fluid, or with a solid 
or a gas, a portion of the energy of the system (that, namely, 
which we call surface energy), is proportional to the area of the 
surface of contact: it is also proportional to a coefficient which 
is specific for each particular pair of substances, and which is 
constant for these, save only in so far as it may be modified by 


294 | ‘THE FORMS OF TISSUES (on. 


changes of temperature or of electric charge. The condition of 
munimum potential energy in the system, which is the condition of 
equilibrium, will accordingly be obtained by the utmost possible 
diminution in the area of the surfaces in contact. When we have 
three bodies in contact, the case becomes a little more complex. 
Suppose for instance we have a drop of some fluid, A, floating on 
another fluid, B, and exposed to air, C. The whole surface energy 
of the system may now be considered as divided into two parts, 
one at the surface of the drop, and the other outside of the same; 
the latter portion is inherent in the surface BC, between the mass 
of fluid B and the superincumbent air, C; but the former portion 
consists of two parts, for it is divided between the two surfaces AB 
and AC, that namely which separates the drop from the surrounding 
fluid and that which separates it from the atmosphere. So far as 


the drop is concerned, then, equilibrium depends on a proper 
balance between the energy, per unit area, which is resident in 
its own two surfaces, and that which is externa! thereto: that is 
to say, if we call #,, the energy at the surface between the two 
fluids, and so on with the other two pairs of surface energies, the 
condition of equilibrium, or of maintenance of the drop, is that - 


By. < Ea its ae 


If, on the other hand, the fluid A happens to be oil and the fluid 
B, water, then the energy per unit area of the water-air surface 
is greater than that of the oil-air surface and that of the oil-water 
surface together ; i.e. 


Egg Hog te ou 


Here there is no equilibrium, and in order to obtain it the water-air 
surface must always tend to decrease and the other two interfacial 
surfaces to increase; which is as much as to say that the water 
tends to become covered by a spreading film of oil, and the water- 
air surface to be abolished. 


vit] OR CELL-AGGREGATES 295 


The surface energy of which we have here spoken is manifested 
in that contractile force, or “tension,” of which we have already 
had so much to say*. In any part of the free water surface, for 
instance, one surface particle attracts another surface particle, and 
the resultant of these multitudinous attractions is an equilibrium 
of tension throughout this particular surface. In the case of our 
three bodies in contact with one another, and within a small area 

very near to the point of contact, a water particle (for instance) 
will be pulled outwards by another water particle; but on the 
opposite side, so to speak, there will be no water surface, and no 
water particle, to furnish the counterbalancing pull; this counter- 


Fig. 100. 


Fig. 101. 


pull, which is necessary for equilibrium, must therefore be provided 
by the tensions existing in the other two surfaces of contact. In 
short, if we could imagine a single particle placed at the very point 
of contact, it would be drawn upon by three different forces, 
whose directions would he in the three surface planes, and whose 
magnitude would be proportional to the specific tensions charac- 
teristic of the two bodies which in each case combine to form the 
“interfacial” surface. Now for three forces acting at a point to 
be in equilibrium, they must be capable of representation, in 
magnitude and direction, by the three sides of a triangle, taken in 
order, in accordance with the elementary theorem of the Triangle 
of Forces. So, if we know the form of our floating drop (Fig. 100), 
then by drawing tangents from O (the point of mutual contact), 

* It can easily be proved (by equating the increase of energy stored in an 


increased surface to the work done ia increasing that surface), that the tension 
measured per unit breadth, 7',,, is equal to the energy per unit area, L,,,. 


296 THE FORMS OF TISSUES [CH. 


we determine the three angles of our triangle (Fig. 101), and we 
therefore know the relative magnitudes of the three surface 
tensions, which magnitudes are proportional to its sides; and 
conversely, if we know the magnitudes, or relative magnitudes. 
of the three sides of the triangle, we also know its angles, and these 
determine the form of the section of the drop. It is scarcely 
necessary to mention that, since all points on the edge of the 
drop are under similar conditions, one with another, the form of 
the drop, as we look down upon it from above, must be circular, . 
and the whole drop must be a solid of revolution. 


The principle of the Triangle of Forces is expanded, as follows, 
by an old seventeenth-century theorem, called Lami’s Theorem : 
“Tf three forces acting at a point be in equilibrium, each force is 
proportional to the sine of the angle contained between the directions 
of the other two.” That is to say 


PO: RR: — sn QOR : sm, POR ssn POG: 


or ee SSC ay wee 
sin QOR sin ROP sin POQ’ 

And from this, in turn, we derive the equivalent formulae, by 
which each force is expressed in terms of the other two, and of the 


angle between them: 
P?2—= Q? + R? + 20R cos (QOR), etc. 


From this and the foregoing, we learn the following important 
and useful deductions: 

(1) The three forces can only be in equilibrium when any one 
of them is less than the sum of the other two: for otherwise, the 
triangle is impossible. Now in the case of a drop of olive-oil 
upon a clean water surface, the relative magnitudes of the three 
tensions (at 15° C.) have been determined as follows: 


Water-air surface Am my. 15 
Oil-air surface ... aks yet 32 
Oil-water surface oie Be: 21 


No triangle having sides of these relative magnitudes is possible ; 
and no such drop therefore can remain in equilibrium. 


vit] OF SACHS’S RULE 297 


(2) The three surfaces may be all alike: as when a soap- 
bubble floats upon soapy water, or when two soap-bubbles are 
joined together, on either side of a partition-film. In this case, 
the three tensions are all equal, and therefore the three angles 
are all equal; that is to say, when three similar liquid surfaces 
meet together, they always do so at an angle of 120°. Whether 
our two conjoined soap-bubbles be equal or unequal, this is still 
the invariable rule; because the specific tension of a particular 
surface is unaffected by any changes of magnitude or form. 

(3) If two only of the surfaces be alike, then two of the 
angles will be alike, and the other will be unlike; and this last 
will be the difference between 360° and the sum of the other two. 
A particular case is when a film is stretched between solid and 
~ parallel walls, like a soap-film within a cylindrical tube. Here, so 
long as there is no external pressure applied to either side, so long 
as both ends of the tube are open or closed, the angles on either 
side of the film will be equal, that is to say the film will set itself 
at right angles to the sides. 

Many years ago Sachs laid it down as a principle, which has 
become celebrated in botany under the name of Sachs’s Rule, 
that one cell-wall always tends to set itself at right angles to another ° 
cell-wall. This rule applies to the case which we have just illus- 
trated; and such validity as the rule possesses is due to the fact 
that among plant-tissues it very frequently happens that one 
cell-wall has become solid and rigid before another and later 
partition-wall is developed in connection with it. 

(4) There is another important principle which arises not out 
of our equations but out of the general considerations by which 
we were led to them. We have seen that, at and near the point 
of contact between our several surfaces, there is a continued 
balance of forces, carried, so to speak, across the interval; in 
other words, there is physical continuity between one surface and 
another. It follows necessarily from this that the surfaces merge 
one ihto another by a continuous curve. Whatever be the form 
of our surfaces and whatever the angle between them, this small 
intervening surface, approximately spherical, is always there to 
bridge over the line of contact* ; and this little fillet, or “ bourrelet,”’ 


* The presence of this little liquid “ bourrelet,”’ drawn from the material of which 


298 THE FORMS OF TISSUES [CH. 


as Plateau called it, is large enough to be a common and con- 
spicuous feature in the microscopy of tissues (Fig. 102). For 
instance, the so-called “splitting” of the cell-wall, which is con- 
spicuous at the angles of the large “ parenchymatous”’ cells in the 
succulent tissues of all higher plants (Fig. 103), is nothing more 
than a manifestation of Plateau’s “bourrelet,” or surface of 
continuity *. 


We may now illustrate some of the foregoing principles, 
before we proceed to the more complex cases in which more 
bodies than three are in mutual contact. But in doing so, we 
must constantly bear in mind the principles set forth in our 
chapter on the forms of cells, and especially those relating to the 
pressure exercised by a curved film. 


Fig. 102. (After Berthold.) Fig. 103. Parenchyma of Maize. 


Let us look for a moment at the case presented by the partition- 
wall in a double soap-bubble. As we have just seen, the three 
films in contact (viz. the outer walls of the two bubbles and the 
partition-wall between) being all composed of the same substance 


the partition-walls themselves are composed, is obviously tending to a reduction 
of the internal surface-area. And it may be that it is as well, or better, accounted 
for on this ground than on Plateau’s assumption that it represents a “surface of 
continuity.” : 

* A similar “bourrelet” is admirably seen at the line of junction between a 
floating bubble and the liquid on which it floats; in which case it constitutes a 
“masse annulaire,” whose mathematical properties and relation to the form of the 
nearly hemispherical bubble, have been investigated by van der Mensbrugghe (cf. 
Plateau, op. cit., p. 386). The form of the superficial vacuoles in Actinophrys or 
Actinosphaerium involves an identical problem. 


vit] OF PLATEAU’S BOURRELET 299 


and all alike in contact with air, the three surface tensions must 
be equal; and the three films must therefore, in all cases, meet 
at an angle of 120°. But, unless the two bubbles be of precisely 
equal size (and therefore of equal curvature) it is obvious that the 
tangents to the spheres will not meet the plane of their circle 
of contact at equal angles, and therefore that the partition-wall 
must be a curved surface: it is only plane when it divides two 
equal and symmetrical cells. It is also obvious, from the sym- 
metry of the figure, that the centres of the spheres, the centre of 
the partition, and the centres of the two spherical surfaces are 
all on one and the same straight line. 

Now the surfaces of the two bubbles exert a pressure inwards 
which is inversely proportional to their radii: that is to say 
p:p::ifr:1/r; and the partition wall must, for equilibrium, 
exert a pressure (P) which is equal to the difference between these 


Fig. 104. 


two pressures, that is to say, P=1/R=1/r’ —1fr=(r—r')/rr’. It 
follows that the curvature of the partition wall must be just such 
a curvature as is capable of exerting this pressure, that is to say, 
R=rr'/(ry —7’). The partition wall, then, is always a portion of 
a spherical surface, whose radius is equal to the product, divided 
by the difference, of the radii of the two vesicles. It follows at 
once from this that if the two bubbles be equal, the radius of 
curvature of the partition is infinitely great, that.1s to say the 
partition is (as we have already seen) a plane surface. 

The geometrical construction by which we obtain the position 
of the centres of the two spheres and also of the partition surface 
is very simple, always provided that the surface tensions are 
uniform throughout the system. If p be a point of contact 
between the two spheres, and cp be a radius of one of them, then 
make the angle cpm = 60°, and mark off on pm, pe’ equal to the 


300 THE FORMS OF TISSUES [CH. 


radius of the other sphere; in like manner, make the angle 
c'pn = 60°, cutting the line cc’ in c’’; then c’ will be the centre 
of the second sphere, and c’’ that of the spherical partition. 
Whether the partition be or be not a plane surface, it is obvious 
that its line of yunction with the rest of the system lies in a plane, 


Fig. 105. ; Fig. 106. 


and is at right angles to the axis of symmetry. The actual 
curvature of the partition-wall is easily seen in optical section; 
but in surface view, the line of junction is projected as a plane 
(Fig. 106), perpendicular to the axis, and this appearance has 
also helped to lend support and authority to “Sachs’s Rule.” 


A @ECOEL CCS Many spherical cells, such as 


Protococcus, divide into two equal 
halves, which are therefore separ- 
COOGOGE5 ated by a plane partition. Among 
the other lower Algae, akin to 


B 
QOS eae, Protococcus, such as the Nostocs 


and Oscillatoriae, in which the 
; cells are imbedded in a gelatinous 
OSTSECE LEGS : 

matrix, we find a series of forms 
such as are represented in Fig. 107. 
D PLL ores Sometimes the cells are solitary 
; or disunited; sometimes they run 

Fig. 107. Filaments, or chains of . ; a : 
cells, in various lower Algae. 10 pal’s OF 1n TOWS, separated one 
(A) Nostoc; (B) Anabaena; (C) 3 pie é 
Rivularia; (D) Oscillatoria. from another by flat partinien= 
and sometimes the conjoined cells 
are approximately hemispherical, but at other times each half 
is more than a hemisphere. These various conditions depend, 


vit] . OF CELL-PARTITIONS 301 


according to what we have already learned, upon the relative 
magnitudes of the tensions at the surface of the cells and at the 
boundary between them®*. 

In the typical case of an equally divided cell, such as a double 
and co-equal soap-bubble, where the partition-wall and the outer 
walls are similar to one another and in contact with similar sub- 
stances, we can easily determine the form of the system. For, at 
any point of the boundary of the partition-wall, O, the tensions 
being equal, the angles QOP, ROP, QOR are all equal, and each 
is, therefore, an angle of 120°. But OQ, OR being tangents, the 
centres of the two spheres (or circular arcs in the figure) lie on 
perpendiculars to them; therefore the radii CO, C’O meet at an 


R 


P 
Fig. 108. 


angle of 60°, and COC’ is an equilateral triangle. That is to say, 
the centre of each circle lies on the circumference of the other; 
the partition lies midway between the two centres; and the 
length (i.e. the diameter) of the partition-wall, PO, is 


2, sin 60° = 1-732 


times the radius, or -866 times the diameter, of each of the cells. 
This gives us, then, the form of an aggregate of two equal cells 
under uniform conditions. 

As soon as the tensions become unequal, whether from changes 
in their own substance or from differences in the substances with 
which they are in contact, then the form alters. If the tension 


* Tn an actual calculation we must of course always take account of the tensions 
on both sides of each film or membrane. 


302 THE FORMS OF TISSUES [CH. 


along the partition, P, diminishes, the partition itself enlarges, - 
and the angle YOR increases: until, when the tension P is very 
small compared to @ or R, the whole figure becomes a circle, and 
the partition-wall, dividing it into two hemispheres, stands at 
right angles to the outer wall. This is the case when the outer 
wall of the cell is practically solid. On the other hand, if P begins 
to increase relatively to Q and R, then the partition-wall contracts, 
and the two adjacent cells become larger and larger segments of 
a sphere, until at length the system becomes divided into two 
separate cells. 

In the spores of Liverworts (such as Pellia), the first partition- 
wall (the equatorial partition in Fig. 109, a) divides the spore into 
- two equal halves, and is therefore a plane surface, normal to the 
surface of the cell; but the next partitions arise near to either 


Fig. 109. Spore of Pellia. (After Campbell.) 


end of the original spherical or elliptical cell. Each of these latter 
partitions will (like the first) tend to set itself normally to the 
cell-wall; at least the angles on either side of the partition will 
be identical, and their magnitude will depend upon the tension 
existing between the cell-wall and the surrounding medium. 
They will only be right angles if the cell-wall is already practically 
solid, and in all probability (rigidity of the cell-wall not being 
quite attained) they will be somewhat greater. In either case 
the partition itself will be a portion of a sphere, whose curvature 
will now denote a difference of pressures in the two chambers or 
cells, which it serves to separate. (The later stages of cell-division, 
represented in the figures b and c, we are not yet in a position to 
deal with.) 

We have innumerable cases, near the tip of a growing filament, 
where in like manner the partition-wall which cuts off the terminal 


ee 


vit] OF CELL-PARTITIONS 303 


cell constitutes a spherical lens-shaped surface, set normally to 
the adjacent walls. At the tips of the branches of many Florideae, 
for instance, we find such a lenticular partition. In Dvzctyota 
dichotoma, as figured by Reinke, we have a succession of such 


‘partitions; and, by the way, in such cases as these, where the 


tissues are very transparent, we have often in optical section a 
puzzling confusion of lines; one being the optical section of the 
curved partition-wall, the other being the straight linear projection 
of its outer edge to which we have already referred. In the 
conical terminal cell of Chara, we have the same lens-shaped 
curve, but a little lower down, where the sides of the shoot are 
approximately parallel, we have flat transverse partitions, at the 
edges of which, however, we recognise a convexity of the outer 
cell-wall and a definite angle of contact, equal on the two sides 


of the partition. 
Q | 
(ere 


Fig. 110. Cells of Dictyota. Fig. 111. Terminal and other cells 
(After Reinke.) ot Chara. 


In the young antheridia of Chara (Fig. 112), and in the not 
dissimilar case of the sporangium (or conidiophore) of Mucor, we 
easily recognise the hemispherical form of the septum which shuts 
off the large spherical cell from the cylindrical 
filament. Here, in the first phase of develop- 
ment, we should have to take into consideration 
the different pressures exerted bys the single 
curvature of the cylinder and the double 
curvature of its spherical cap (p. 221); and 
we should find that the partition would have 
a somewhat low curvature, with a radius less 
than the diameter of the cylinder; which it 
would have exactly equalled but for the Fig. 112. Young 


“4: : : : : antheridium of 
additional pressure inwards which it receives Chara. 


304 THE FORMS OF TISSUES [cH. 


from the curvature of the large surrounding sphere. But as the 
latter continues to grow, its curvature decreases, and so likewise 
does the inward pressure of its surface; and accordingly the little 
convex partition brlges out more and more. 


In order to epitomise the foregoing facts let the annexed 
diagrams (Fig. 113) represent a system of three films, of which 
one is a partition-wall between the other two; and let the tensions 
at the three surfaces, or the tractions exercised upon a point at 
their meeting-place, be proportional to T, T’ and ¢t. Let a, B, y 
be, as in the figure, the opposite angles. Then: 

(1) If T be equal to T’, and ¢ be relatively insignificant, 
the angles a, 8 will be of 90°. 


fi yv a Reg T! 
ae ale 
ib t 
Fig. 113. 


(2) If T= T’, but be a little greater than ¢, then ¢ will exert 
an appreciable traction, and a, 8 will be more than 90°, say, for 
instance, 100°. 

(3) IfiT=T’ =i, then a, 8, y will all equal 120°. 

The more complicated cases, when ¢, T and T’ are all unequal, 
are already sufficiently explained. 


The biological facts which the foregoing considerations go a 
long way to explain and account for have been the subject of much 
argument and discussion, especially on the part of the botanists. 
Let me recapitulate, in a very few words, the history of this long 
discussion. 

Some fifty years ago, Hofmeister laid it down as a general law 
that “The partition-wall stands always perpendicular to what was 
previously the principal direction of growth in the cell,”—or, in 
other words, perpendicular to the long axis of the cell*. Ten 


* Hofmeister, Pringsheim’s Jahrb. 11, p. 272, 1863; Hdb. d. physiol. Bot. 1, 
1867, p. 129. 


vit] OF CELL-PARTITIONS 305 


years later, Sachs formulated his rule, or principle, of “rectangular 
section,” declaring that in all tissues, however complex, the 
cell-walls cut one another (at the time of their formation) at right 
angles*. Years before, Schwendener had found, in the final 
results of cell-division, a universal system of “orthogonal tra- 
jectoriest”’; and this idea Sachs further developed, introducing 
complicated systems of confocal ellipses and hyperbole, and 
distinguishing between periclinal walls, whose curves approximate 
to the peripheral contours, radial partitions, which cut these at 
an angle of 90°, and finally anticlines, which stand at right angles 
to the other two. 

Reinke, in 1880, was the first to throw some doubt upon this 
explanation. He pointed out various cases where the angle was 
not a right angle, but was very definitely an acute one; and 
he saw, apparently, in the more common rectangular symmetry 
merely what he calls a necessary, but secondary, result of growth. 

Within the next few years, a number of botanical writers were 
content to point out further exceptions to Sachs’s Rule§; and in 
some cases to show that the curvatures of the partition-walls, 
especially such cases of lenticular curvature as we have described, 
were by no means accounted for by either Hofmeister or Sachs; 
while within the same period, Sachs himself, and also Rauber, 
attempted to extend the main generalisation to animal tissues). 

While these writers regarded the form and arrangement of the 
cell-walls as a biological phenomenon, with little if any direct 
relation to ordinary physical laws, or with but a vague reference 
to ‘“‘mechanical conditions,’ the physical side of the case was 
soon urged by others, with more or less force and cogency. Indeed 
the general resemblance between a cellular tissue and a “froth” 


* Sachs, Ueber die Anordnung der Zellen in jiingsten Pflanzentheilen, Verh. 
phys. med. Ges. Wiirzburg, x1, pp. 219-242, 1877; Ueber Zellenanordnung und 
Wachsthum, zbid. xu, 1878; Ueber die durch Wachsthum bedingte Verschiebung 
kleinster Theilchen in trajectorischen Curven, Monatsber. k. Akad. Wiss. Berlin, 
1880; Physiology of Plants, chap. xxvii, pp. 431-459, Oxford, 1887. 

+ Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus, 
Naturf. Ges. Ziirich, Febr. 1860, pp. 272-296. 

{t Reinke, Lehrbuch der Botanik, 1880, p. 519. 

§ Cf. Leitgeb, Unters. iiber die Lebermoose, u, p. 4, Graz, 1881. 

|| Rauber, Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. v1, 
pp. 279, 334, 1882. 


Gia 20 


306 THE FORMS OF TISSUES [CH. 


had been pointed out long before, by Melsens, who had made an 
“artificial tissue” by blowing into a solution of white of egg*. 

In 1886, Berthold published his Protoplasmamechantk, in which 
he definitely adopted the principle of “minimal areas,” and, 
following on the lines of Plateau, compared the forms of many 
cell-surfaces and the arrangement of their partitions with those 
assumed under surface tension by a system of “ weightless films.” 
But, as Klebsf points out in reviewing Berthold’s book, Berthold 
was careful to stop short of attributing the biological phenomena 
to a definite mechanical cause. They remained for him, as they 
had done for Sachs, so many “phenomena of growth,” or 
“properties of protoplasm.” 

In the same year, but while still apparently unacquainted with 
Berthold’s work, Errerat published a short but very lucid article, 
in which he definitely ascribed to the cell-wall (as Hofmeister had 
already done) the properties of a semi-liquid film and drew from 
this as a logical consequence the deduction that it must assume the 
various configurations which the law of minimal areas imposes on 
the soap-bubble. So what we may call Errera’s Law is formulated 
as follows: A cellular membrane, at the moment of its formation, 
tends to assume the form which would be assumed, under the 
same conditions, by a liquid film destitute of weight. 

Soon afterwards Chabry, in discussing the embryology of the 
Ascidians, indicated many of the points in which the contacts 
between cells repeat the surface-tension phenomena of the soap- 
bubble, and came to the conclusion that part, at least, of the 
embryological phenomena were purely physical§; and the same 
line of investigation and thought were pursued and developed by 
Robert, in connection with the embryology of the Mollusca|]. 
Driesch again, in a series of papers, continued to draw attention 
to the presence of capillary phenomena in the segmenting cells 


* O. R. Acad. Sc. xxxuu, p. 247, 1851; Ann. de chimie et de phys. (3), XXXUI, 
p- 170, 1851; Bull. R. Acad. Belg. xxtv, p. 531, 1857. 

+ Klebs, Biolog. Centralbl. vu, pp. 193-201, 1°87. 

{ L. Errera, Sur une condition fondamentale d’équilibre des cellules vivantes, 
CO. R., cm, p. 822, 1886; Bull. Soc. Belge de Microscopie, xm, Oct. 1886; Recueil 
Peres (Phy stole générale), 1910, pp. 201-205. 

§ L. Chabry, Embryologie des Ascidiens, J. Anat. et Physiol. xxun, p. 266, 1887. 

| Robert, Embryologie des Troques, Arch. de Zool. exp. et gén. (3), X, 1892. 


vit] OF ERRERA’S LAW 307 


of various embryos, and came to the conclusion that the mode of 
segmentation was of little importance as regards the final result*. 

Lastly de Wildemant, in a somewhat wider, but also vaguer 
generalisation than Errera’s, declared that “The form of the 
cellular framework of vegetables, and also of animals, in its 
essential features, depends upon the forces of molecular physics.” 


Let us return to our problem of the arrangement of partition 
films. When we have three bubbles in contact, instead of two as 
in the case already considered, the phenomenon 1s strictly analogous 
to our former case. The three bubbles will be separated by three 
partition surfaces, whose curvature will depend upon the relative 


Fig. 114. 


size of the spheres, and which will be plane if the latter are all of 
the same dimensions; but whether plane or curved, the three 
partitions will meet one another at an angle of 120°, in an axial 
line. Various pretty geometrical corollaries accompany this ar- 
rangement. For instance, if Fig. 114 represent the three associated 
bubbles in a plane drawn through their centres, ¢, c’, c’’ (or what 
is the same thing, if it represent the base of three bubbles resting 
on a plane), then the lines wc, ue”, or sc, sc’, etc., drawn to the 


* “Dass der Furchungsmodus etwas fiir das Zukiinftige unwesentliches ist,” 
Z. f. w. Z. wv, 1893, p. 37. With this statement compare, or contrast, that of 
Conklin, quoted on p. 4; cf. also pp. 157, 348 (footnotes). 

+ de Wildeman, Etudes sur l’attache des cloisons cellulaires, Mém. Couwronn. 
del Acad. R. de Belgique, t111, 84 pp., 1893-4. 


20— 2 


308 THE FORMS OF TISSUES [oH. 


centres from the points of intersection of the circular arcs, will 
_always enclose an angle of 60°. Again (Fig. 115), if we make the 
angle c”uf equal to 60°, and produce uf to meet cc” in f, f will be 
the centre of the circular arc. which constitutes the partition Ow; 
and further, the three points f,g, h, successively determined in this 


' 
4 
‘ 1 


- 


e2ts- 


Fig. 115. 


manner, will lie on one and the same straight line. In the case 


of coequal bubbles or cells (as in Fig. 114, B), it is obvious that 
the lines joining their centres form an equilateral triangle; and 
consequently, that the centre of each circle (or sphere) lies on the 
circumference of the other two; it is also obvious that uf is now 


vu] OF LAMARLE’S LAW 309 


parallel to cc”, and accordingly that the centre of curvature of 
the partition is now infinitely distant, or (as we have already said), 
that the partition itself is plane. 

When we have four bubbles in conjunction, they would seem 
to be capable of arrangement in two symmetrical ways: either, 
as in Fig. 116 (A), with the four partition-walls meeting at right 
angles, or, as in (B), with five partitions meeting, three and three, 
at angles of 120°. This latter arrangement is strictly analogous 
to the arrangement of three bubbles in Fig. 114. Now, though 
both of these figures, from their symmetry, are apparently figures of 
equilibrium, yet, physically, the former turns out to be of unstable 


A B 
Fig. 116. 


and the latter of stable equilibrium. If we try to bring our four 
bubbles into the form of Fig. 116, A, such an arrangement endures 
only for an instant; the partitions glide upon each other, a median 
wall springs into existence, and the system at once assumes the 
form of our second figure (B). This is a direct consequence of the 
law of minimal areas: for it can be shewn, by somewhat difficult 
mathematics (as was first done by Lamarle), that, in dividing a 
closed space into a given number of chambers by means of partition- 
walls, the least possible area of these partition-walls, taken together, 
can only be attained when they meet together in groups of three, 
at equal angles, that is to say at angles of 120°. 


310 THE FORMS OF TISSUES [cH. 


Wherever we have a true cellular complex, an arrangement of 
cells in actual physical contact by means of a boundary film, we 
find this general principle in force; we must only bear in mind 
that, for its perfect recognition, we must be able to view the 
object in a plane at right angles to the boundary walls. For 
instance, in any ordinary section of a vegetable parenchyma, we 
recognise the appearance of a “froth,” ‘precisely resembling that 
which we can construct by imprisoning a mass of soap-bubbles in 
a narrow vessel with flat sides of glass; in both cases we see the 
cell-walls everywhere meeting, by threes, at angles of 120°, irre- 
spective of the size of the individual cells: whose relative size, on 
the other hand, determines the curvature of the partition-walls. 
On the surface of a honey-comb we have precisely the same 
conjunction, between cell and cell, of three boundary walls, 
meeting at 120°. In embryology, when we examine a segmenting 
egg, of four (or more) segments, we find in like manner, in the great 
majority of cases, if not in all, that the same principle is still 
exemplified; the four segments do not meet in a common centre, 
but each cell is in contact with two others, and the three, and only 
three, common boundary walls meet at the normal angle of 120°. 
A so-called polar furrow*, the visible edge of a vertical partition- 
wall, joins (or separates) the two triple contacts, precisely as in 
Fig. 116, B. 

In the four-celled stage of the frog’s egg, Rauber (an exception- 
ally careful observer) shews us three alternative modes in which 
the four cells may be found to be conjoined (Fig. 117). In (A) we 
have the commonest arrangement, which is that which we have 
just studied and found to be the simplest theoretical one; that 
namely where a straight “polar furrow” intervenes, and where, 
at its extremities, the partition-walls are conjoined three by three. 
In (B), we have again a polar furrow, which is now seen to be a 
portion of the first “segmentation-furrow” (cf. Fig. 155 ete.) by 
which the egg was originally divided into two; the four-celled 
stage being reached by the appearance of the transverse furrows 

* It was so termed by Conklin in 1897, in his paper on Crepidula (J. of Morph. 
xi, 1897). It is the Querfurche of Rabl (Morph. Jahrb. v, 1879); the Polarfurche 
of O. Hertwig (Jen. Zeitschr. xiv, 1880); the Brechungslinie of Rauber (Neue 


Grundlage zur K. der Zelle, M. Jb. vim, 1882). It is carefully discussed by Robert, 
Dév. des Troques, Arch. de Zool. Exp. et Gén. (3), X, 1892, p. 307 seq. 


vit] OF THE POLAR FURROW 311 


and their corresponding partitions. In this case, the polar 
furrow is seen to be sinuously curved, and Rauber tells us that 
its curvature gradually alters: as a matter of fact, it (or rather 
the partition-wall corresponding to it) is gradually setting itself 
into a position of equilibrium, that is to say of equiangular contact 
with its neighbours, which position of equilibrium is already 
attained or nearly so in Fig. 117, A. In Fig. 117, C, we have a 
very different condition, with which we shall deal in a moment. 

According to the relative magnitude of the bodies in contact, 
this “polar furrow” may be longer or shorter, and it may be so 
minute as to be not easily discernible; but it is quite certain that 
no simple and homogeneous system of fluid films such as we 
are dealing with is in equilibrium without its presence. In the 
accounts given, however, by embryologists of the segmentation of 
the egg, while the polar furrow is depicted in the great majority 


A 


Fig. 117. Various ways in which the four cells are co-arranged in 
the four-celled stage of the frog’s egg. (After Rauber.) 


of cases, there are others in which it has not been seen and some 
in which its absence is definitely asserted*. The cases where four 
cells, lying in one plane, meet a a point, such as were frequently 
figured by the older embryologists, are very difficult to verify, 
and I have not come across a single clear case in recent literature. 
Considering the physical stability of the other arrangement, the 
great preponderance of cases in which it is known to occur, the 
difficulty of recognising the polar furrow in cases where it is 
very small and unless it be specially looked for, and the natural 
tendency of the draughtsman to make an all but symmetrical 
structure appear wholly so, I am much inclined to attribute to 


* Thus Wilson (J. of Morph. vit, 1895) declared that in Amphioxus the polar 
furrow was occasionally absent, and Driesch took occasion to criticise and to throw 
doubt upon the statement (Arch. f. Hntw. Mech. 1, 1895, p. 418). 


312 THE FORMS OF TISSUES [CH. 


error or imperfect observation all those cases where the junction- 
lines of four cells are represented (after the manner of Fig. 116, A) 
as a simple cross*. 

But while a true four-rayed intersection, or simple cross, is 
theoretically impossible (save as a transitory and highly unstable 
condition), there is another condition which may closely simulate 
it, and which is common enough. There are plenty of repre- 
sentations of segmenting eggs, in which, instead of the triple 
junction and polar furrow, the four cells (and in like manner their 
more numerous successors) are represented as rounded off, and 
separated from one another by an empty space, or by a little drop 
of an extraneous fluid, evidently not directly miscible with the 
fluid surfaces of the cells. Such is the case in the obviously 
accurate figure which Rauber gives (Fig. 117, C) of the third mode 
of conjunction in the four-celled stage of the frog’s egg. Here 
Rauber is most careful to point out that the furrows do not simply 
“cross,” or meet in a point, but are separated by a little space, 
which he calls the Polgriibchen, and asserts to be constantly present 
whensoever the polar furrow, or Brechungslinie, is not to be 
discerned. This little interposed space, with its contained drop 
of fluid, materially alters the case, and implies a new condition 
of theoretical and actual equilibrium. For, on the one hand, we 
see that now the four intercellular partitions do not meet one 
another at all; but really impinge upon four new and separate 
partitions, which constitute interfacial contacts, not between cell 
and cell, but between the respective cells and the intercalated 
drop. And secondly, the angles at which these four little surfaces 
will meet the four cell-partitions, will be determined, in the usual 
way, by the balance between the respective tensions of these several 
surfaces. In an extreme case (as in some pollen-grains) it may be 
found that the cells under the observed circumstances are not truly 
in surface contact: that they are so many drops which touch but 
do not “wet” one another, and which are merely held together 
by the pressure of the surrounding envelope. But even supposing, 

* Precisely the same remark was made long ago by Driesch: “Das so oft 
schematisch gezeichnete Vierzellenstadium mit zwei sich in zwei Punkten scheidende 
Medianen kann man wohl getrost aus der Reihe des Existierenden streichen,” 


Entw. mech. Studien, Z. f. w. Z. Lm, p. 166, 1892. Cf. also his Math. mechanische 
Bedeutung morphologischer Probleme der Biologie, Jena, 59 pp. 1891. 


vi] OF THE POLAR FURROW 313 


as 1s in all probability the actual case, that they are in actual fluid 
contact, the case from the point of view of surface tension presents 
no difficulty. In the case of the conjoined soap-bubbles, we were 
dealing with semilar contacts and with equal surface tensions through- 
out the system; but in the system of protoplasmic cells which 
constitute the segmenting egg we must make allowance for an in- 
equality of tensions, between the surfaces where cell meets cell, and 
where on the other hand cell-surface is in contact with the sur- 
rounding medium,—ain this case generally water or one of the fluids 
of the body. Remember that our general condition is that, in our 
entire system, the sum of the surface energies is a minimum; and, 
while this is attained by the sum 
of the surfaces being a minimum 
in the case where the energy is 
uniformly distributed, it is not 
necessarily so under non-uniform 
conditions. In the diagram (Fig. 
118) if the energy per unit area 
be greater along the contact 
surface cc’, where cell meets cell, 
than along ca or cb, where cell- 
surface is in contact with the surrounding medium, these latter 
surfaces will tend to increase and the surface of cell-contact 
to diminish. In short there will be the usual balance of forces 
between the tension along the surface cc’, and the two opposing 
tensions along ca and cb. If the former be greater than either 
of the other two, the outside angle will be less than 120°; and if 
the tension along the surface cc’ be as much or more than the 
sum of the other two, then the drops will stand in contact only, 
save for the possible effect of external pressure, at a point. This is 
the explanation, in general terms, of the peculiar conditions 
obtaining in Nostoc and its allies (p. 300), and it also leads us to 
a consideration of the general properties and characters of an 
“epidermal” layer. 


fol? 
Fig. 118. 


While the inner cells of the honey-comb are symmetrically 
situated, sharing with their neighbours in equally distributed 
pressures or tensions, and therefore all tending with great accuracy 


314 THE FORMS OF TISSUES [CH. 


to identity of form, the case is obviously different with the cells 
at the borders of the system. So it is, in hke manner, with our 
froth of soap-bubbles. The bubbles, or cells, in the interior of 
the mass are all alike in general character, and if they be equal 
in size are alike in every respect: their sides are uniformly 
flattened*, and tend to meet at equal angles of 120°. But the 
bubbles which constitute the outer layer retain their spherical 
surfaces, which however still tend to meet the partition-walls 
connected with them at constant angles of 120°. This outer layer 
of bubbles, which forms the surface of our froth, constitutes after 
a fashion what we should call in botany an “epidermal” layer. 
But in our froth of soap-bubbles we have, as a rule, the same kind 
of contact (that is to say, contact with air) both within and without 
the bubbles; while in our living cell, the outer wall of the epidermal 
cell is exposed to air on the one side, but is in contact with the 


Fig. 119. 


protoplasm of the cell on the other: and this involves a difference 
of tensions, so that the outer walls and their adjacent partitions 
are no longer likely to meet at equal angles of 120°. Moreover, 
a chemical change, due for instance to oxidation or possibly also 
to adsorption, is very likely to affect the external wall, and may 
tend to its consolidation; and this process, as we have seen, is 
tantamount to a large increase, and at the same time an 
equalisation, of tension in that outer wall, and will lead the 
adjacent partitions to impinge upon it at angles more and 
more nearly approximating to 90°: the bubble-like, or spherical, 
surfaces of the individual cells beg more and more flattened 
in consequence. Lastly, the chemical changes which affect the 
outer walls of the superficial cells may extend, in greater or 
less degree, to their inner walls also: with the result that these 


* Compare, however, p. 299. 


vit] OF EPIDERMAL TISSUES 315 


cells will tend to become more or less rectangular throughout, and 
will cease to dovetail into the interstices of the next subjacent 
layer. These then are the general characters which we recognise 
in an epidermis; and we perceive that the fundamental character 
of an epidermis simply is that it les on the outside, and that its 
“main physical characteristics follow, as a matter of course, from 
the position which it occupies and from the various consequences 
which that situation entails. We have however by no means 
exhausted the subject in this short account; for the botanist is 
accustomed to draw a sharp distinction between a true epidermis 
and what is called epidermal tissue. The latter, which is found in 
such a sea-weed as Laminaria and in very many other cryptogamic 
plants, consists, as in the hypothetical case we have described, 
of a more or less simple and direct modification of the general or 
fundamental tissue. But a “true epidermis,” such as we have it 
in the higher plants, is something with a long morphological history, 
something which has been laid down or differentiated in an early 
stage of the plant’s growth, and which afterwards retains its 
separate and independent character. We shall see presently that 
a physical reason is again at hand to account, under certain 
circumstances, for the early partitioning off, from a mass of 
embryonic tissue, of an outer layer of cells which from their first 
appearance are marked off from the rest by their rectangular and 
flattened form. 


We have hitherto considered our cells, or bubbles, as lying in 
a plane of symmetry, and further, we have only considered the 
appearance which they present as projected on that plane: in 
simpler words, we have been considering their appearance in 
surface or in sectional view. But we have further to consider 
them as solids, whether they be still grouped in relation to a single 
plane (like the four cells in Fig. 116) or heaped upon one another, 
as for instance in a tetrahedral form like four cannon-balls; and in 
either case we have to pass from the problems of plane to those of 
solid geometry. In short, the further development of our theme 
must lead us along two paths of enquiry, which continually 
intercross, namely (1) the study of more complex cases of partition 
and of contact in a plane, and (2) the whole question of the surfaces 


316 THE FORMS OF TISSUES © [CH. 


and angles presented by solid figures in symmetrical juxtaposition. 
Let us take a simple case of the latter kind, and again afterwards, 
so far as possible, let us try to keep the two themes separate. 
Where we have three spheres in contact, as in Fig. 114 or in 
either half of Fig. 116, B, let us consider the point of contact 
(O, Fig. 114) not as a point in the plane section of the diagram, but 
as a point where three furrows meet on the surface of the system. 
At this point, three cells meet; but it is also obvious that there meet 
here six surfaces, namely the outer, spherical walls of the three 
bubbles, and the three partition-walls which divide them, two and 
two. Also, four lines or edges meet here; viz. the three external arcs 
which form the outer boundaries of the partition-walls (and which 
correspond to what we commonly call the “furrows” in the seg- 
menting egg); and as a fourth edge, the “arris” or junction of the 
three partitions (perpendicular to the plane of the paper), where 
they all three meet together, as we have seen, at equal angles of 
120°. Lastly, there meet at the point fowr solid angles, each 
bounded by three surfaces: to wit, within each bubble a solid 
angle bounded by two partition-walls and by the surface wall; 
and (fourthly) an external solid angle bounded by the outer 
surfaces of all three bubbles. Now in the case of the soap-bubbles 
(whose surfaces are all in contact with air, both outside and in), 
the six films meeting at the point, whether surface films or partition 
films, are all similar, with similar tensions. In other words the 
tensions, or forces, acting at the point are all similar and symmet- 
rically arranged, and it at once follows from this that the angles, 
solid as well as plane, are all equal. It is also obvious that, as 
regards the point of contact, the system will still be symmetrical, 
and its symmetry will be quite unchanged, if we add a fourth 
bubble in contact with the other three: that is to say, 1f where 
we had merely the outer air before, we now replace it by the air 
in the interior of another bubble. The only difference will be that 
the pressure exercised by the walls of this fourth bubble will alter 
the curvature of the surfaces of the others, so far as it encloses 
them; and, if all four bubbles be identical in size, these surfaces 
which formerly we called external and which have now come to 
be internal partitions, will, like the others, be flattened by equal 
and opposite pressure, into planes. We are now dealing, in short, 


vit] OF TETRAHEDRAL SYMMETRY alT 


with six planes, meeting symmetrically in a point, and constituting 
there four equal solid angles. 

If we make a wire cage, in the form of a regular tetrahedron, 
and dip it into soap-solution, then when we withdraw it we see 
that to each one of the six edges of the tetrahedron, i.e. to each 
one of the six wires which constitute the little cage, a film has 
attached itself; and these six films meet internally at a point, and 
constitute in every respect the symmetrical figure which we have 
_ just been describing. In short, the system of films we have 
hereby automatically produced is precisely the system of partition- 
walls which exist in our tetrahedral aggregation of four spherical 


Qa 


b Ste 1 
Cc 
Fig. 120. 


bubbles :—precisely the same, that is to say, in the neighbourhood 
of the meeting-point, and only differing in that we have made the 
wires of our tetrahedron straight, instead of imitating the circular 
arcs which actually form the intersections of our bubbles. ‘This 
detail we can easily introduce in our wire model if we please. 
Let us look for a moment at the geometry of our figure. Leto 
(Fig. 120) be the centre of the tetrahedron, i.e. the centre of sym- 
metry where our films meet; and let oa, ob, oc, od, be lines drawn to 
the four corners of the tetrahedron. Produce ao to meet the base 
in p; then apd is a right-angled triangle. It is not difficult to 
prove that in such a figure, 0 (the centre of gravity of the system) 


318 THE FORMS OF TISSUES [CH. 


lies just three-quarters of the way between an apex, a, and a point, 
p, which is the centre of gravity of the opposite base. Therefore 


op = oa/3 = od/3. 
Therefore cos dop = 4, and cos aod = — }. 


That is to say, the angle aod is just, as nearly as possible, 
109° 28’ 16’. This angle, then, of 109° 28’ 16’, or very nearly 
109 degrees and a half, is the angle at which, in this and every 
other solid system of liquid films, the edges of the partition-walls 
meet one another at a point. It is the fundamental angle in the 
solid geometry of our systems, just as 120° was the fundamental 
angle of symmetry so long as we considered only the plane pro- 
jection, or plane section, of three films meeting in an edge. 


Out of these two angles, we may construct a great variety of 
figures, plane and solid, which become all the more varied and 
complex when, by considering the case of unequal as well as equal 
cells, we admit curved (e.g. spherical) as well as plane boundary 
surfaces. Let us consider some examples and illustrations of 
these, beginning with those which we need only consider in reference 
to a plane. 

Let us imagine a system of equal cylinders, or equal spheres, 
in contact with one another in a plane, and represented in section 
by the equal and contiguous circles of Fig. 121. I borrow my 
figure, by the way, from an old Italian naturalist, Bonanni (a. 
contemporary of Borelh, of Ray and Willoughby and of Martin 
Lister), who dealt with this matter in a book chiefly devoted to 
molluscan shells*. 

It is obvious, as a simple geometrical fact, that each of these 
equal circles is in contact with six surrounding circles. Imagine 
now that the whole system comes under some uniform stress. 
It may be of uniform surface tension at the boundaries of all the 
cells; it may be of pressure caused by uniform growth or expansion 
within the cells; or it may be due to some uniformly applied 
constricting pressure from without. In all of these cases the points 
of contact between the circles in the diagram will be extended into 


* Ricreatione dell’ occhio e della mente, nell’ Osservatione delle Chiocciole, Roma, 
1681. 


vit] OF HEXAGONAL SYMMETRY 319 


lines of contact, representing surfaces of contact in the actual 
spheres or cylinders; and the equal circles of our diagram will 
be converted into regular and equal hexagons. The angles of 
these hexagons, at each of which three hexagons meet, are of 
course angles of 120°. So far as the form is concerned, so long as 
we are concerned only with a morphological result and not with 
a physiological process, the result is precisely the same whatever 
be the force which brings the bodies together in symmetrical 
apposition ; it is by no means necessary for us, in the first instance, 
even to enquire whether it be surface tension or mechanical 


Fig. 121. Diagram of hexagonal cells. (After Bonanni.) 


pressure or some other physical force which is the cause, or the 
main cause, of the phenomenon. 

The production by mutual interaction of polyhedral cells, 
which, under conditions of perfect symmetry, become regular 
hexagons, is very beautifully illustrated by Prof. Bénard’s 
“tourbillons cellulaires”’ (cf. p. 259), and also in some of Leduc’s 
diffusion experiments. A weak (5 per cent.) solution of gelatine 
is allowed to set on a plate of glass, and little drops of a 5 or 
10 per cent. solution of ferrocyanide of potassium are then placed 
at regular intervals upon the gelatine. Immediately each little 
drop becomes the centre, or pole, of a system of diffusion currents, 


320 THE FORMS OF TISSUES [CH. 


and the several systems conflict with and repel one another, so 
that presently each little area becomes the seat of a double current 
system, from its centre outwards and back again; until at length 
the concentration of the field becomes equalised and the currents 


Fig. 122. An “artificial tissue,” formed by coloured drops of sodium chloride 
solution diffusing in a less dense solution of the same salt. (After Leduc.) 


Fig. 123. An artificial cellular tissue, formed by the diffusion in gelatine of 
drops of a solution of potassium ferrocyanide. (After Leduc.) 


————————E~ OSS 


vit] OF HEXAGONAL SYMMETRY 321 


cease. After equilibrium is attained, and when the gelatinous 
mass is permitted to dry, we have an artificial tissue of more or 
less regularly hexagonal “cells,” which simulate in the closest way 
an organic parenchyma. And by varying the experiment, in ways 
which Leduc describes, we may simulate various forms of tissue, 
and produce cells with thick walls or with thin, cells in close 
contact or with wide intercellular spaces, cells with plane or with 
curved partitions, and so forth. 


The hexagonal pattern is illustrated among organisms in count- 
less cases, but those in which the pattern is perfectly regular, by 
reason of perfect uniformity of force and perfect equality of the 
individual cells, are not so numerous. The hexagonal epithelium- 
cells of the pigment layer of the eye, external to the retina, are 
a good example. Here we have a single layer of uniform cells, 
reposing on the one hand upon a basement membrane, supported 


“a 


Fig. 124. Epidermis of Girardia. (After Goebel.) 


behind by the solid wall of the sclerotic, and exposed on the other 
hand to the uniform fluid pressure of the vitreous humour. The 
conditions all point, and lead, to a perfectly symmetrical result : 
that is to say, the cells, uniform in size, are flattened out to a 
uniform thickness by the fluid pressure acting radially; and their 
reaction on each other converts the flattened discs into regular 
hexagons. In an ordinary columnar epithelium, such as that of 
the intestine, we see again that the columnar cells have been 
compressed into hexagonal prisms; but here as a rule the cells 
are less uniform in size, small cells are apt to be intercalated 
among the larger, and the perfect symmetry is accordingly lost. 
The same is true of ordinary vegetable parenchyma; the originally 
spherical cells are approximately equal in size, but only approxi- 
mately ; and there are accordingly all degrees in the regularity and 
symmetry of the resulting tissue. But obviously, wherever we 


Ti (eh = ll 


322 THE FORMS OF TISSUES [CH. 


have, in addition to the forces which tend to produce the regular 
hexagonal symmetry, some other asymmetrical component arising 
from growth or traction, then our regular hexagons will be dis- 
torted in various simple ways. This condition is illustrated in 
the accompanying diagram of the epidermis of Girardia; it also 
accounts for the more or less pointed or fusiform cells, each still 
in contact (as a rule) with six others, which form the epithelial 
lining of the blood-vessels: and other similar, or analogous, 
instances are very common. 

In a soap-froth imprisoned between two glass plates, we have 
a symmetrical system of cells, which appear in optical section (as 


Fig. 125. Soap-froth under pressure. (After Rhumbler.) 


in Fig. 125, B) as regular hexagons; but if we press the plates a 

little closer together, the hexagons become deformed or flattened 
(Fig. 125, A). In this case, however, if 
we cease to apply further pressure, the 
tension of the films throughout the 
system soon adjusts itself again, and in a 
short time the system has regained the 
former symmetry of Fig. 125, B. 

In the growth of an ordinary dicoty- 
ledonous leaf, we once more see reflected in 
the form of its epidermal cells the tractions, 
irregular but on the whole longitudinal, 

Fig. 126. From leaf of Which growth has superposed on the ten- 
Elodea canadensis. (After sions of the partition-walls (Fig. 126). In 
el) the narrow elongated leaf of a Monocoty- 

ledon, such as a hyacinth, the elongated, apparently quadrangular 


vit] OF HEXAGONAL SYMMETRY 323 


cells of the epidermis appear as a necessary consequence of the 
simpler laws of growth which gave its simple form to the leaf as 
a whole. In this last case, however, as in all the others, the rule 
still holds that only three partitions (in surface view) meet in a 
point; and at their point of meeting the walls are for a short 
distance manifestly curved, so as to permit the junction to take 
place at or nearly at the normal angle of 120°. 

Briefly speaking, wherever we have a system of cylinders or 
spheres, associated together with sufficient mutual interaction to 
bring them into complete surface contact, there, in section or in 
surface view, we tend to get a pattern of hexagons. 


While the formation of an hexagonal pattern on the basis of ready-formed 
and symmetrically arranged material units is a very common, and indeed the 
general way, it does not follow that there are not others by which such a 
pattern can be obtained. For instance, if we take a little triangular dish of 
mercury and set it vibrating (either by help of a tuning-fork, or by simply 
tapping on the sides) we shall have a series of little waves or ripples starting 
inwards from each of the three faces; and the intercrossing, or interference 
of these three sets of waves produces crests and hollows, and intermediate 
points of no disturbance, whose loci are seen as a beautiful pattern of minute 
hexagons. It is possible that the very minute and astonishingly regular 
pattern of hexagons which we see, for instance, on the surface of many diatoms, 
may be a phenomenon of this order*. The same may be the case also in Arcella, 
where an apparently hexagonal pattern is found not to consist of simple 
hexagons, but of “straight lines in three sets of parallels, the lines of each 
set making an angle of sixty degrees with those of the other two sets{.” We 
must also bear in mind, in the case of the minuter forms, the large possibilities 
of optical illusion. For instance, in one of Abbe’s “‘diffraction-plates,” a 
pattern of dots, set at equal interspaces, is reproduced on a very minute scale 
by photography; but under certain conditions of microscopic illumination 
and focussing, these isolated dots appear as a pattern of hexagons. 


ce 


A symmetrical arrangement of hexagons, such as we have just been 
studying, suggests various simple geometrical corollaries, of which the following 
may perhaps be a useful one. 

We may sometimes desire to estimate the number of hexagonal areas or 
facets in some structure where these are numerous, such for instance as the 


* Cf. some of J. H. Vincent’s photographs of ripples, in Phil. Mag. 1897-1899; 
or those of F. R. Watson, in Phys. Review, 1897, 1901, 1916. The appearance will 
depend on the rate of the wave, and in turn on the surface-tension; with a low 
tension one would probably see only a moving “jabble.’’ FitzGerald thought 
diatom-patterns might be due to electromagnetic vibrations (Works, p. 503, 1902). 

+ Cushman, J. A. and Henderson, W. P., Amer. Nat. xu, pp. 797-802, 1906. 

21—2 


324 THE FORMS OF TISSUES [CH 


cornea of an insect’s eye, or in the minute pattern of hexagons on many diatoms. 
An approximate enumeration is easily made as follows. 

For the area of a hexagon (if we call 6 the short diameter, that namely 
which bisects two of the opposite sides) is 62 x 3/2, the area of a circle 
being d?. 7/4. Then, if the diameter (d) of a circular area include n hexagons, 
the area of that circle equals (n. 5)? x 7/4. And, dividing this number by 
the area of a single hexagon, we obtain for the number of areas in the circle, ' 
each equal to a hexagonal facet, the expression n2 x 2/4 x 2/s/3 = 0-907n2, or 
9/10. ?, nearly. 

This calculation deals, not only with the complete facets, but with the 
areas of the broken hexagons at the periphery of the circle. If we neglect 
these latter, and consider our whole field as consisting of successive rings of 
hexagons about a central one, we may obtain a still simpler rule*. For 
obviously, around our central hexagon there stands a zone of six, and around 
these a zone of twelve, and around these a zone of eighteen, and soon. And 
the total number, excluding the central hexagon, is accordingly: 


For one zone 6 — 2 xeo i en eNO 
», two zones 18 213) (0 = Dios 
,, three zones 36 — Ag =) oe a 
,, four zones 60 = 15) oI = aoc hoe iy 
,, five zones 90 =6§x 15 = 3 x5 x65 


and so forth. If NV be the number of zones, and if we add one to the above 
numbers for the odd central hexagon, the rule evidently is, that the total 
number, H, = 3N(N+1)+ 1. Thus, if in a preparation of a fly’s cornea, 
I can count twenty-five facets in a line from a central one, the total number 
in the entire circular field is (3 x 25 x 26) + 1= 195I1f. 


The same principles which account for the development of 
hexagonal symmetry hold true, as a matter of course, not only 
of individual cells (in the biological sense), but of any close- 
packed bodies of uniform size and originally circular outline; 
and the hexagonal pattern is therefore of very common occurrence, 
under widely different circumstances. The curious reader may 
consult Sir Thomas Browne’s quaint and beautiful account, in the 
Garden of Cyrus, of hexagonal (and also of quincuncial) symmetry 
in plants and animals, which “doth neatly declare how nature 
Geometrizeth, and observeth order in all things.” 


* This does not merely neglect the broken ones but all whose centres lie between 
this circle and a hexagon inscribed in it. 

+ For more detailed calculations see a paper by “H.M.” [? H. Munro], in 
Q. J. M.S. vi, p. 83, 1858. 


vit] OF HEXAGONAL SYMMETRY 325 


*We have many varied examples of this principle among corals, 
wherever the polypes are in close juxtaposition, with neither 
empty space nor accumulations of matrix between their adjacent 
walls. Favosites gothlandica, for instance, furnishes us with an 
excellent example. In the great genus Lithostrotion we have some 
species that are “massive” and others that are “fasciculate” ; in 
other words in some the long cylindrical corallites are in close con- 
tact with one another, and in others they are separate and loosely 
bundled (Fig. 127). Accordingly in the former the corallites are 


Fig. 127. Lithostrotion Martini. Fig. 128. Cyathophyllum hexagonum. 
(After Nicholson.) (From Nicholson, after Zittel.) 


% 


squeezed into hexagonal prisms, while in the latter they retain their 
cylindrical form. Where the polypes are comparatively few, and 
so have room to spread, the mutual pressure ceases to work or 
only tends to push them asunder, letting them remain circular in 
outline (e.g. Thecosmila). Where they vary gradually in size, as 
for instance in Cyathophyllum hexagonum, they are more or less 
hexagonal but are not regular hexagons; and where there is greater 
and more irregular variation in size, the cells will be on the 
average hexagonal, but some will have fewer and some more sides 
than six, as in the annexed figure of Arachnophyllum (Fig. 129). 


326 THE FORMS OF TISSUES [CH. 


Where larger and smaller cells, corresponding to two different 
kinds of zooids, are mixed together, we may get various results. 
If the larger cells are numerous enough to be more or less in 
contact with one another (e.g. various Monticuliporae) they will 
be irregular hexagons, while the smaller cells between them will 
be crushed into all manner of irregular angular forms. If on the 
other hand the large cells are comparatively few and are large 
and strong-walled compared with their smaller neighbours, then 
the latter alone will be squeezed into hexagons, while the larger 
ones will tend to retain their circular outline undisturbed (e.g. 
Heliopora, Heliolites, etc.). 

When, as happens in certain corals,.the peripheral walls or 


Fig. 129. Arachnophyllum pentagonum. Fig. 130. Heliolites. (After 
(After Nicholson.) Woods.) 


“thecae” of the individual polypes remain undeveloped but 
the radiating septa are formed and calcified, then we obtain new 
and beautiful mathematical configurations (Fig. 131). For the 
radiating septa are no longer confined to the circular or hexagonal 
bounds of a polypite, but tend to meet and become confluent 
with their neighbours on every side; and, tending to assume 
positions of equilibrium, or of minimal area, under the restraints 
to which they are subject, they fall into congruent curves; and 
these correspond, in a striking manner, to the lines of force running, 
in a common field of force, between a number of secondary centres. 
Similar patterns may be produced in various ways, by the play 
of osmotic or magnetic forces; and a particular and very curious 
case is to be found in those complicated forms of nuclear division 


vit] OF HEXAGONAL SYMMETRY 327 


known as triasters, polyasters, etc., whose relation to a field of 
force Hartog has explained*. It is obvious that, in our corals, 
these curving septa are all orthogonal to the non-existent hexagonal 
boundaries. As the phenomenon is wholly due to the imperfect 
development or non-existence of a thecal wall, it is not surprising 
that we find identical configurations among various corals, or 
families of corals, not otherwise related to one another; we find 
the same or very similar patterns displayed, for instance, in 
Synhelia (Oculinidae), in Phillipsastraea (Rugosa), in Thamnas- 
traea (Fungida), and in many more. 


The most famous of all hexagonal conformations and perhaps 
the most beautiful is that of the bee’s cell. Here we have, as in 


Fig. 131. Surface-views of Corals with undeveloped thecae and confluent septa. 
A, Thamnastraea; B, Comoseris. (From Nicholson, after Zittel.) 


our last examples, a series of equal cylinders, compressed by 
symmetrical forces into regular hexagonal prisms. But in this 
case we have two rows of such cylinders, set opposite to one 
another, end to end; and we have accordingly to consider also 
the conformation of their ends. We may suppose our original 
cylindrical cells to have spherical ends, which is their normal and 
symmetrical mode of termination; and, for closest packing, it is 
obvious that the end of any one cylinder will touch, and fit in 
between, the ends of three cylinders in the opposite row. It is 
just as when we pile round-shot in a heap; each.sphere that we 


* Cf. Hartog, The Dual Force of the Dividing Cell, Science Progress (n.s.), I, 
Oct. 1907, and other papers. Also Baltzer, Ueber mehrpolige Mitosen bei Seeigel- 
eiern, Inaug. Diss. 1908. 


328 THE FORMS OF TISSUES [cH. 


set down fits into its nest between three others, and the four 
form a regular tetrahedral arrangement. Just as it was obvious, 
then, that by mutual pressure from the six laterally adjacent cells, 
any one cell would be squeezed into a hexagonal prism, so is it also 
obvious that, by mutual pressure against the three terminal 
neighbours, the end of any one cell will be compressed into a solid 
trihedral angle whose edges will meet, as in the analogous case 
already described of a system of soap-bubbles, at a plane angle 
of 109° and so many minutes and.seconds. What we have to 
comprehend, then, is how the scx sides of the cell are to be combined 
with its three terminal facets. This is done by bevelling off three 
alternate angles of the prism, in a uniform manner, until we have 
tapered the prism to a point; and by so doing, we evidently 
produce three rhombic surfaces, each of which is double of the 
triangle formed by joining the apex to the three untouched angles 
of the prism. If we experiment, not with cylinders, but with 
spheres, if for instance we pile together a mass of bread-pills (or 
pills of plasticine), and then submit the whole to a uniform pressure, 
it 1s obvious that each ball (like the seeds in a pomegranate, as 
Kepler said), will be in contact with twelve others,—six in its own 
plane, three below and three above, and in compression it will 
therefore develop twelve plane surfaces. It will in short repeat, 
above and below, the conditions to which the bee’s cell is subject 
at one end only; and, since the sphere is symmetrically situated 
towards its neighbours on all sides, it follows that the twelve plane 
sides to which its surface has been reduced will be all similar, 
equal and similarly situated. Moreover, since we have produced 
this result by squeezing our original spheres close together, it is 
evident that the bodies so formed completely fill space. The 
regular solid which fulfils all these conditions is the rhombic 
dodecahedron. The bee’s cell, then, is this figure incompletely 
formed: it is a hexagonal prism with one open or unfinished end, 
and one trihedral apex of a rhombic dodecahedron. 

The geometrical form of the bee’s cell must have attracted the 
attention and excited the admiration of mathematicians from time 
immemorial. Pappus the Alexandrine has left us (in the intro- 
duction to the Fifth Book of his Collections) an account of its 
hexagonal plan, and he drew from its mathematical symmetry the 


vit] OF THE BEE’S CELL 329 


conclusion that the bees were endowed with reason: “There 
being, then, three figures which of themselves can fill up the 
space round a point, viz. the triangle, the square and the hexagon, 
the bees have wisely selected for their structure that which contains 
most angles, suspecting indeed that it could hold more honey than 
either of the other two.” Erasmus Bartholinus was apparently 
the first to suggest that this hypothesis was not warranted, and 
that the hexagonal form was no more than the necessary result 
of equal pressures, each bee striving to make its own little circle 
as large as possible. 

The investigation of the ends of the cell was a more difficult 
matter, and came later, than that of its sides. In general terms 
this arrangement was doubtless often studied and described: as 
for instance, in the Garden of Cyrus! “And the Combes them- 
selves so regularly contrived that their mutual intersections 
make three Lozenges at the bottom of every Cell; which severally 
regarded make three Rows of neat Rhomboidall Figures, connected 
at the angles, and so continue three several chains throughout the 
whole comb.” But Maraldi* (Cassini’s nephew) was the first to 
measure the terminal solid angle or determine the form of the 
rhombs in the pyramidal ending of the cell. He tells us that the 
angles of the rhomb are 110° and 70°: “Chaque base d’alvéole 
est formée par trois rhombes presque toujours égaux et semblables, 
qui, suivant les mesures que nous avons prises, ont les deux angles 
obtus chacun de 110 degrés, et par conséquent les deux aigus 
chacun de 70°.” He also stated that the angles of the trapeziums 
which form the sides of the body of the cell were identical angles, 
of 110° and 70°; but in the same paper he speaks of the angles as 
being, respectively, 109° 28’ and 70° 32’. Here a singular con- 
fusion at once arose, and has been perpetuated in the booksy. 
“Unfortunately Réaumur chose to look upon this second deter- 
mination of Maraldi’s as being, as well as the first, a direct result 
of measurement, whereas it is in reality theoretical. He speaks of 
it as Maraldi’s more precise measurement, and this error has been 
repeated in spite of its absurdity to the present day; nobody 


* Observations sur les Abeilles, Mém. Acad. Sc. Paris, 1712, p. 299. 

+ As explained by Leslie Ellis, in his essay “On the Form of Bees’ Cells,” 
in Mathematical and other Writings, 1853, p. 353; cf. O. Terquem, Nouv. Ann. 
Math. 1856, p. 178. 


330 THE FORMS OF TISSUES [CH. 


appears to have thought of the impossibility of measuring such a 
thing as the end of a bee’s cell to the nearest minute.” At any 
rate, it now occurred to Réaumur (as curiously enough, it had not 
done to Maraldi) that, just as the closely packed hexagons gave 
the minimal extent of boundary in a plane, so the actual solid 
figure, as determined by Maraldi, might be that which, for a given 
solid content, gives the minimum of surface: or which, in other 
words, would hold the most honey for the least wax. He set this 
problem before Koenig, and the geometer confirmed his conjecture, 
the result of his calculations agreeing within two minutes (109° 26’ 
and 70° 34’) with Maraldi’s determination. But again, Maclaurin* 
and Lhuiliert, by different, methods, obtained a result identical 
with Maraldi’s; and were able to shew that the discrepancy of 
2’ was due to an error in’ Koenig’s calculation (of tan 6= 4/2), 
—that is to say to the imper- 
fection of his logarithmic tables,— 
not (as the books say{) “to a 
mistake on the part of the Bee.” 
“Not to a mistake on the part of 
Maraldi”’ is, of course, all that we 
are entitled to say. 

The theorem may be proved as 
follows : 

ABCDEF, abcdef, is a right 
prism upona regular hexagonal base. 
The corners BDF are cut off by 
planes through the lines AC, CE, 
EA, meeting in a point V on the 
axis VN of the prism, and intersect- 
ing Bb, Dd, Ff, at X, Y, Z. Th is 
evident that the volume of the figure 
thus formed is the same as that of 
the original prism with hexagonal 
ends. For, if the axis cut the 
hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. 


Fig. 132. 


* Phil. Trans. xuu, 1743, pp. 565-571. + Mém. del Acad. de Berlin, 1781. 
t Cf. Gregory, Examples, p. 106, Wood’s Homes without Hands, 1865, p. 428, 
Mach, Science of Mechanics, 1902, p. 453, etc., ete. 


Vit] - OF THE BEER’S CELL 331 


It is required to find the inclination of the faces forming the 
trihedral angle at V to the axis, such that the surface of the 
figure may be a minimum. 

Let the angle NV X, which is half the solid angle of the prism, 
==19*) the side of the hexagon, as AB, =a; and the height, as 
Aa, = h. 

Then, AC = 2a cos 30° = a+/3. 

And VX = a/sin @ (from inspection of the triangle LX B) 

Therefore the area of the rhombus VAXC = a? 4/3/2 sin 6. 

And the area of AabX = a/2 (2h — $VX cos 6) 

= a/2 (2h — a/2.cot 6). 


Therefore the total area of the figure 


a ara/3 
= hexagon abcdef + 3a (20 — 5 cot 0) +3 Fan" 


d(Area) 3a? / 1 4/3 cos 6 
Tpcrctore Ro vinnie? Ga sin? 6 ) ; 
But this expression vanishes, that is to say, d (Area)/dé = 0, 
when cos 6 = 1/4/3, that is when 6 = 54° 44’ 8” 
a OO 28) 1G! 7s 
This then is the condition under which the total area of the 
figure has its minimal value. 


That the beautiful regularity of the bee’s architecture is due 
to some automatic play of the physical forces, and that it were 
fantastic to assume (with Pappus and Réaumur) that the bee 
intentionally seeks for a method of economising wax, is certain, 
but the precise manner of this automatic action is not so clear. 
When the hive-bee builds a solitary cell, or a small cluster of cells, 
as 1t does for those eggs which are to develop into queens, it makes 
but a rude production. The queen-cells are lumps of coarse wax 
hollowed out and roughly bitten into shape, bearing the marks of 
the bee’s jaws, like the marks of a blunt adze on a rough-hewn log. 
Omitting the simplest of all cases, when (as among some humble- 
bees) the old cocoons are used to hold honey, the cells built by 
the “solitary”? wasps and bees are of various kinds. They may 
be formed by partitioning off little chambers in a hollow stem; 


332 THE FORMS OF TISSUES [CH. 


they may be rounded or oval capsules, often very neatly con- 
structed, out of mud, or vegetable fibre or little stones, agglutinated 
together with a salivary glue; but they shew, except for their 
rounded or tubular form, no mathematical symmetry. The social 
wasps and many bees build, usually out of vegetable matter 
chewed into a paste with saliva, very beautiful nests of “combs” ; 
and the close-set papery cells which constitute these combs are 
just as regularly hexagonal as are the waxen cells of the hive-bee. 
But in these cases (or nearly all of them) the cells are in a single 
row; their sides are regularly hexagonal, but their ends, from the 
want of opponent forces, remain simply spherical. In Melipona 
domestica (of which Darwin epitomises Pierre Huber’s description) 
“the large waxen honey-cells are nearly spherical, nearly equal in 
size, and are aggregated into an irregular mass.’ But the spherical 
form is only seen on the outside of the mass; for inwardly each 
cell is flattened into “two, three or more flat surfaces, according 
as the cell adjoins two, three or more other cells. When one cell 
rests on three other cells, which from the spheres being nearly 
of the same size is very frequently and necessarily the case, the 
three flat surfaces are united into a pyramid; and this pyramid, as 
Huber has remarked, is manifestly a gross imitation of the three- 
sided pyramidal base of the cell of the hive-bee*.”? The question 
is, to what particular force are we to ascribe the plane surfaces 
and definite angles which define the sides of the cell in all these 
cases, and the ends of the cell in cases where one row meets and 
opposes another. We have seen that Bartholin suggested, and it 
is still commonly believed, that this result is due to simple physical 
pressure, each bee enlarging as much as it can the cell which it 
is a-building, and nudging its wall outwards till it fills every 
intervening gap and presses hard against the similar efforts of 
its neighbour in the cell next doory. But it is very doubtful 

* Origin of Species, ch. vur (6th ed., p. 221). The cells of various bees, 
humble-bees and social wasps have been described and mathematically investigated 
by K. Miillenhoff, Pfliiger’s Archiv xxxu, p. 589, 1883; but his many interesting 
results are too complex to epitomise. For figures of various nests and combs see 
(e.g.) von Biittel-Reepen, Biol. Centralbl. xxxu1, pp. 4, 89, 129, 183, 1903. 

+ Darwin had a somewhat similar idea, though he allowed more play to the 
bee’s instinct or conscious intention. Thus, when he noticed certain half-completed 


cell-walls to be concave on one side and convex on the other, but to become perfectly 
flat when restored for a short time to the hive, he says: “It was absolutely im- 


vu] OF THE BEE’S CELL 333 


whether such physical or mechanical pressure, more or less inter- 
mittently exercised, could produce the all but perfectly smooth, 
plane surfaces and the all but perfectly definite and constant 
angles which characterise the cell, whether it be constructed of 
wax or papery pulp. It seems more likely that we have to do 
with a true surface-tension effect; in other words, that the walls 
assume their configuration when in a semi-fluid state, while the 
papery pulp is still liquid, or while the wax is warm under the 
high temperature of the crowded hive*. Under these circum- 
stances, the direct efforts of the wasp or bee may be supposed 
to be limited to the making of a tubular cell, as thin as the nature 
of the material permits, and packing these little cells as close as 
possible together. It is then easily conceivable that the sym- 
metrical tensions of the adjacent films (though somewhat retarded 
by viscosity) should suffice to bring the whole system into equi- 
librium, that is to say into the precise configuration which the 
comb actually presents. In short, the Maraldi pyramids which 
terminate the bee’s cell are precisely identical with the facets of 
a rhombic dodecahedron, such as we have assumed to constitute 
(and which doubtless under certain conditions do constitute) the 
surfaces of contact in the interior of a mass of soap-bubbles or 
of uniform parenchymatous cells; and there is every reason to 
believe that the physical explanation is identical, and not merely 
mathematically analogous. 

The remarkable passage in which Buffon discusses the bee’s 
cell and the hexagonal configuration in general is of such historical 
importance, and tallies so closely with the whole trend of our 
enquiry, that I will quote it in full: “Dirai-je encore un mot; 
ces cellules des abeilles, tant vantées, tant admirées, me fournissent 
une preuve de plus contre l’enthousiasme et admiration; cette 
figure, toute géométrique et toute réguliére qu’elle nous parait, et 
qu'elle est en effet dans la spéculation, n’est ici qu’un résultat 
mécanique et assez imparfait qui se trouve souvent dans la nature, 
possible, from the extreme thinness of the little plate, that they could have effected 
this by gnawing away the convex side; and I suspect that the bees in such cases 
stand on opposite sides and push and bend the ductile and warm wax (which as 
I have tried is easily done) into its proper intermediate piane, and thus flatten it.” 


* Since writing the above, I see that Miillenhoff gives the same explanation, 
and declares that the waxen wall is actually a Fliissigkeitshdutchen, or liquid film. 


334 THE FORMS OF TISSUES [CH. 


et que l’on remarque méme dans les productions les plus brutes; 
les cristaux et plusieurs autres pierres, quelques sels, etc., prennent 
constamment cette figure dans leur formation. Qu’on observe les 
petites écailles de la peau d’une roussette, on verra qu’elles sont 
hexagones, parce que chaque écaille croissant en méme temps se 
fait obstacle, et tend 4 occuper le plus d’espace qu'il est possible 
dans un espace donné: on voit ces mémes hexagones dans le 
second estomac des animaux ruminans, on les trouve dans les 
graines, dans leurs capsules, dans certaines fleurs, etc. Qu’on 
remplisse un vaisseau de pois, ou plitot de quelque autre graine 
cylindrique, et qu’on le ferme exactement aprés y avoir versé 
autant d’eau que les intervalles qui restent entre ces graines 
peuvent en recevoir; qu’on fasse bouillir cette eau, tous ces 
cylindres deviendront de colonnes a six pans*. On y voit claire- 
ment la raison, qui est purement mécanique; chaque graine, dont 
la figure est cylindrique, tend par son renflement a occuper le 
plus d’espace possible dans un espace donné, elles deviennent done 
toutes nécessairement hexagones par la compression réciproque. 
Chaque abeille cherche a occuper de méme le plus d’espace possible 
dans un espace donné, il est donc nécessaire aussi. puisque le 
corps des abeilles est cylindrique, que leurs cellules sont hexagones, 
—par la méme raison des obstacles réciproques. On donne plus 
(esprit aux mouches dont les ouvrages sont les plus réguliers ; 
les abeilles sont, dit-on, plus ingénieuses que les guépes, que les 
frélons, etc., qui savent aussi larchitecture, mais dont les con- 
structions sont plus grossiéres et plus irréguliéres que celles des 
abeilles: on ne veut pas voir, ou l’on ne se doute pas que cette 
régularité, plus ou moins grande, dépend uniquement du nombre 
et de la figure, et nullement de lintelligence de ces petites bétes ; 
plus elles sont nombreuses, plus il y a des forces qui agissent 
également et s’opposent de méme, plus il y a par conséquent de 
contrainte mécanique, de régularité forcée, et de perfection 
apparente dans leurs productions 7.” 


* Bonnet criticised Buffon’s explanation, on the ground that his description 
was incomplete; for Buffon took no account of the Maraldi pyramids. 

+ Buffon, Histoire Naturelle, tv, p. 99. Among many other papers on the 
Bee’s cell, see Barclay, Mem. Wernerian Soc. 11, p. 259 (1812), 1818; Sharpe, Phil. 
Mag. tv, 1828, pp. 19-21; L. Lalanne, Ann. Sci. Nat. (2) Zool. xu, pp. 358-374, 
1840; Haughton, Ann. Mag. Nat Hist. (3), x1, pp. 415-429, 1863; A. R. Wallace. 


vit] OF HEXAGONAL SYMMETRY 335 


A very beautiful hexagonal symmetry, as seen in section, or 
dodecahedral, as viewed in the solid, is presented by the cells 
which form the pith of certain rushes (e.g. Juncus effusus), and 
somewhat less diagrammatically by those’ which make the pith 
of the banana. These cells are stellate in form, and the tissue 
presents in section the appearance of a network of six-rayed 
stars (Fig. 133, c), inked together by the tips of the rays, and 
separated by symmetrical, air-filled, intercellular spaces. In thick 
sections, the solid twelve-rayed stars may be very beautifully seen 
under the binocular microscope. 


Fig. 133. Diagram of development of “stellate cells,’ in pith of Juncus. 
(The dark, or shaded, areas represent the cells; the light areas being the 
gradually enlarging “intercellular spaces.’’) 


What has happened here is not difficult to understand. 
Imagine, as before, a system of equal spheres all in contact, each 
one therefore touching six others in an equatorial plane; and let 
the cells be not only in contact, but become attached at the points 
of contact. Then instead of each cell expanding, so as to encroach 
on and fill up the intercellular spaces, let each cell tend to contract 
or shrivel up, by the withdrawal of fluid from its interior. The 


ibid. x1, p. 303, 1863; Jeffries Wyman, Pr. Amer. Acad. of Arts and Sc. vit, pp. 
68-83, 1868; Chauncey Wright, zbid. tv, p. 432, 1860. 


336 THE FORMS OF TISSUES [cH. 


result will obviously be that the intercellular spaces will increase ; 
the six equatorial attachments of each cell (Fig. 133, a) (or its twelve 
attachments in all, to adjacent cells) will remain fixed, and the 
portions of cell-wall between these points of attachment will be 
withdrawn in a symmetrical fashion (b) towards the centre. As 
the final result (c) we shall have a “dodecahedral star” or star- 
polygon, which appears in section as a six-rayed figure. It is 
obviously necessary that the pith-cells should not only be attached 
to one another, but that the outermost layer should be firmly 
attached to a boundary wall, so as to preserve the symmetry of 
the system. What actually occurs in the rush is tantamount to- 
this, but not absolutely identical. Here it is not so much the 
pith-cells. which tend to shrivel within a boundary of constant 
size, but rather the boundary wall (that is, the peripheral ring of 
woody and other tissues) which continues to expand after the 
pith-cells which it encloses have ceased to grow or to multiply. 
The twelve points of attachment on the spherical surface of each 
little pith-cell are uniformly drawn asunder; but the content, or 
volume, of the cell does not increase correspondingly; and the 
remaining portions of the surface, accordingly, shrink inwards and 
gradually constitute the complicated surface of a twelve-pointed 
star, which is still a symmetrical figure and is still also a surface 
of minimal area under the new conditions. . 


A few years after the publication of Plateau’s book, Lord 
Kelvin shewed, in a short but very beautiful paper*, that we must 
not hastily assume from such arguments as the foregoing, that 
a close-packed assemblage of rhombic dodecahedra will be the true 
and general solution of the problem of dividing space with a 
minimum partitional area, or will be present in a cellular liquid 
“foam,” in which it is manifest that the problem is actually and 
automatically solved. The general mathematical solution of the 
problem (as we have already indicated) is, that every interface or 
partition-wall must have constant curvature throughout; that 
where such partitions meet in an edge, they must intersect at 
angles such that equal forces, in planes perpendicular to the line 


* Sir W. Thomson, On the Division of Space with Minimum Partitional Area, 
Phil. Mag. (5), xxiv, pp. 503-514, Dec. 1887; .cf. Baltimore Lectures, 1904, p. 615. 


vit] OF THE PARTITIONING OF SPACE 337 


of intersection, shall balance; and finally, that no more than three 
such interfaces may meet in a line or edge, whence it follows that 
the angle of intersection of the film-surfaces must be exactly 120°. 
An assemblage of equal and similar rhombic dodecahedra goes far 
to meet the case: it completely fills up space; all its surfaces or 
interfaces are planes, that is to say, surfaces of constant curvature 
throughout; and these surfaces all meet together at angles of 120°. 
Nevertheless, the proof that our rhombic dodecahedron (such as 
we find exemplified in the bee’s cell) is a surface of minimal area, 
is not a comprehensive proof; it is limited to certain conditions, 
and practically amounts to no more than this, that of the regular 
solids, with all sides plane and similar, this one has the least surface 
for its solid content. 

The rhombic dodecahedron has six tetrahedral angles, and 
eight trihedral angles; and it is obvious, on consideration, that 
at each of the former six dodecahedra meet in a point, and that, 
where the four tetrahedral facets of each coalesce with their 
neighbours, we have twelve plane films, or interfaces, meeting in 
a point. In a precisely similar fashion, we may imagine twelve 
plane films, drawn inwards from the twelve edges of a cube, to 
meet at a point in the centre of the cube. But, as Plateau dis- 
covered*, when we dip a cubical 
wire skeleton into soap-solution and 
take it out again, the twelve films 
which are thus generated do not 
meet in a point, but are grouped 
around a small central, plane, quadri- 
lateral film (Fig. 134). In other 
words, twelve plane films, meeting in 
a point, are essentially unstable. If 
we blow upon our artificial film- 
system, the little quadrilateral alters 
its place, setting itself parallel now to one and now to another of 
the paired faces of the cube; but we never get rid of it. Moreover, 
the size and shape of the quadrilateral, as of all the other films in the 
system, are perfectly definite. Of the twelve films (which we had 


* Also discovered independently by Sir David Brewster, Trans. R.S.E. XXxtv, 
p- 505, 1867, xxv, p. 115, 1869. 
99) 


T. G. aie 


338 THE FORMS OF TISSUES [CH. 


expected to find all plane and all similar) four are plane isosceles 
triangles, and eight are slightly curved quadrilateral figures. The 
former have two curved sides, meeting at an angle of 109° 28’, 
and their apices coincide with the corners of the central quadri- 
lateral, whose sides are also curved, and also meet at this identical 
angle ;-which (as we observe) is likewise an angle which we have 
been dealing with in the simpler case of the bee’s cell, and indeed 
in all the regular solids of which we have yet treated. 

By completing the assemblage of polyhedra of which Plateau’s 
skeleton-cube gives a part, Lord Kelvin shewed that we should 
obtain a set of equal and similar fourteen-sided figures, or “tetra- 
kaidecahedra”; and that by means of an assemblage of these 
figures space is homogeneously partitioned—that is to say, into 
equal, similar and similarly situated cells—with an economy of 
surface in relation to area even greater than in an assemblage of 
rhombic dodecahedra. 

In the most generalised case, the tetrakaidecahedron is bounded 
by three pairs of equal and parallel quadrilateral faces, and four 
pairs of equal and parallel hexagonal faces, neither the quadri- 
laterals nor the hexagons being necessarily plane. In a certain 
particular case, the quadrilaterals are plane surfaces, but the 
hexagons slightly curved “anticlastic” surfaces; and these latter 
have at every point equal and opposite curvatures, and are 
surfaces of minimal curvature for a boundary of six curved edges. 
The figure has the remarkable property that, like the plane 
rhombic dodecahedron, it so partitions space that three faces 
meeting in an edge do so everywhere at equal angles of 120°*. 

We may take it as certain that, in a system of perfectly fluid 
films, like the interior of a mass of soap-bubbles, where the films 
are perfectly free to glide or to rotate over one another, the mass 
is actually divided into cells of this remarkable conformation. 


* Von Fedorow had already described (in Russian) the same figure, under the 
name of cubo-octahedron, or hepta-parallelohedron, limited however to the case 
where all the faces are plane. This figure, together with the cube, the hexagonal 
prism, the rhombic dodecahedron and the “elongated dodecahedron,” constituted 
the five plane-faced, parallel-sided figures by which space is capable of being 
completely filled and symmetrically partitioned; the series so forming the founda- 
tion of Von Fedorow’s theory of crystalline structure. The elongated dodecahedron 
is, essentially, the figure of the bee’s cell. 


¢ 


vit] OF THE PARTITIONING OF SPACE 339 


_ And it is quite possible, also, that in the cells of a vegetable 
parenchyma, by carefully macerating them apart, the same con- 
formation may yet be demonstrated under suitable conditions ; 

_ that is to say when the whole tissue is highly symmetrical, and the 

individual cells are as nearly as possible equal in size. But in an 
ordinary microscopic section, it would seem practically impossible 
to distinguish the fourteen-sided figure from the twelve-sided. 
Moreover, if we have anything whatsoever interposed so as to 
prevent our twelve films meeting in a point, and (so to speak) to 
take the place of our little central quadrilateral,—if we have, for 
instance, a tiny bead or droplet in the centre of our artificial 
system, or even a little thickening, or “ bourrelet”’ as Plateau called 
it, of the cell-wall, then it is no longer necessary that the 
tetrakaidecahedron should be formed. Accordingly, it is very 
probably the case that, in the parenchymatous tissue, under the 
actual conditions of restraint and of very imperfect fluidity, it is 
after all the rhombic dodecahedral configuration which, even under 
perfectly symmetrical conditions, is generally assumed. 


It follows from all that we have said, that the problems 
connected with the conformation of cells, and with the manner in 
which a given space is partitioned by them, soon become exceedingly 
complex. And while this is so even when all our cells are equal 
and symmetrically placed, it becomes vastly more so when cells 
varying even slightly in size, in hardness, rigidity or other qualities, 
are packed together. The mathematics of the case very soon 
become too hard for us; but in its essence, the phenomenon 
remains the same. We have little reason to doubt, and no just 
cause to disbelieve, that the whole configuration, for instanceg of 
an egg in the advanced stages of segmentation, is accurately 
determined by simple physical laws, just as much as in the early 
stages of two or four cells, during which early stages we are able to 
recognise and demonstrate the forces and their resultant effects. 
But when mathematical investigation has become too difficult, it 
often happens that physical experiment can reproduce for us the 
phenomena which Nature exhibits to us, and which we are striving 

to comprehend. For instance, in an admirable research, M. Robert 
shewed, some years ago, not only that the early segmentation of 


22—2 


340 THE FORMS OF TISSUES [cH. 


the egg of Trochus (a marine univalve mollusc) proceeded in 
accordance with the laws of surface tension, but he also succeeded 
in imitating by means of soap-bubbles, several stages, one after 
another, of the developing egg. 

M. Robert carried his experiments as far as the stage of 
sixteen cells, or bubbles. It is not easy to carry the artificial 
system quite so far, but in the earlier stages the experiment is 
easy; we have merely to blow our bubbles in a little dish, adding 
one to another, and adjusting their sizes to produce a symmetrical 
system. One of the simplest and prettiest parts of his investigation 
concerned the “polar furrow” of which we have spoken on p. 310. 
On blowing four little contiguous bubbles he found (as we may 
all find with the greatest ease) that they form a symmetrical system, 
two in contact with one another by a laminar film, and two, 
which are elevated a little above the others, and which are separated 
by the length of the aforesaid lamina. The bubbles are thus in 
contact three by three, their partition-walls making with one 
another equal angles of 120°. The upper and lower edges of the 
intermediate lamina (the lower one visible through the transparent 
system) constitute the two polar furrows of the embryologist 
(Fig. 135, 1-3). The lamina itself is plane when the system is 
symmetrical, but it responds by a corresponding curvature to 
the least inequality of the bubbles on either side. In the 
experiment, the upper polar furrow is usually a little shorter 
than the lower, but parallel to it; that is to say, the lamina 
is of trapezoidal form: this lack of perfect symmetry being 
due (in the experimental case) to the lower portion of the 
bubbles being somewhat drawn asunder by the tension of their ’ 
ateachments to the sides of the dish (Fig. 135, 4). A similar 
phenomenon is usually found in Trochus, according to Robert, 
and many other observers have likewise found the upper furrow 
to be shorter than the one below. In the various species of the 
genus Crepidula, Conklin asserts that the two furrows are equal 
in C. convexa, that the upper one is the shorter in C. fornicata, 
and that the upper one all but disappears in C. plana; but we may 
well be permitted to doubt, without the evidence of very special 
investigations, whether these slight physical differences are 
actually characteristic of, and constant in, particular allied species. 


vit] OF POLAR FURROWS 341 


Returning to the experimental case, Robert found that by with- 
drawing a little air from, and so diminishing the bulk of the two 
terminal bubbles (i.e. those at the ends of the intermediate lamina), 
the upper polar furrow was caused to elongate, till it became equal 
in length to the lower; and by continuing the process it became 
the longer in its turn. These two conditions have again been 


Fig. 135. Aggregations of four soap-bubbles, to shew various arrangements of 
the intermediate partition and polar furrows. (After Robert.) 


described by investigators as characteristic of this embryo or that; 
for instance in Unio, Lillie has described the two furrows as 
gradually altering their respective lengths*; and Wilson (as Lillie 
remarks) had already pointed out that “the reduction of the 
apical cross-furrow, as compared with that at the vegetative pole 


* F.R. Lillie, Embryology of the Unionidae, Journ. of Morphology, x, p. 12, 
1895. 


342 THE FORMS OF TISSUES [cH. 


in molluscs and annelids ‘stands in obvious relation to the different 
size of the cells produced at the two poles*.’”’ 

When the two lateral bubbles are gradually reduced in size, 
or the two terminal ones enlarged, the upper furrow becomes 
shorter and shorter; and at the moment when it is about to 
vanish, a new furrow makes its instantaneous appearance In a 
direction perpendicular to the old one; but the inferior furrow, 
constrained by its attachment to the base, remains unchanged, 
and accordingly our two polar furrows, which were formerly 
parallel, are now at right angles to one another. Instead of a 
single plane quadrilateral partition, we have now two triangular 
ones, meeting in the middle of the system by their apices, and 
lying in planes at right angles to one another (Fig. 135, 5-7)T. 
Two such polar furrows, equal in length and arranged in a cross, 
have again been frequently described by the embryologists. 
Robert himself found this condition in Trochus, as an occasional 
or exceptional occurrence: it has been described as normal in 
Asterina by Ludwig, in Branchipus by Spangenberg, and in 
Podocoryne and Hydractinia by Bunting. It is evident that it~ 
represents a state of unstable equilibrium, only to be maintained 
under certain conditions of restraint within the system. 

So, by slight and delicate modifications in the relative size of 
the cells, we may pass through all the possible arrangements of the 
median pk&rtition, and of the “furrows” which correspond to its 
upper and lower edges; and every one of these arrangements has 
been frequently observed in the four-celled stage of various embryos. 
As the phases pass one into the other, they are accompanied by 
changes in the curvature of the partition, which in like manner 
correspond precisely to phenomena which the embryologists have 
witnessed and described. And all these configurations belong to 
that large class of phenomena whose distribution among embryos, 
or among organisms in general, bears no relation to the boundaries 
of zoological classification; through molluscs, worms, coelenter- 


* E. B. Wilson, The Cell-lineage of Nereis, Journ. of Morphology, v1, p. 452, 
1892. 

+ It is highly probable, and we may reasonably assume, that the two little 
triangles do not actually meet at an apical point, but merge into one another by 
a twist, or minute surface of complex curvature, so as not to contravene the normal 
conditions of equilibrium. 


vit] OF POLAR FURROWS 343 


ates, vertebrates and what not, we meet with now one and now 
another, in a medley which defies classification. They are not 
“vital phenomena,” or “functions” of the organism, or special 
characteristics of this or that organism, but purely physical 
phenomena. The kindred but more complicated phenomena 
which correspond to the polar furrow when a larger number of 
cells than four are associated together, we shall deal with in the 
next chapter. 

Having shewn that the capillary phenomena are patent and 
unmistakable during the earlier stages of embryonic development, 
but soon become more obscure and incapable of experimental 
reproduction in the later stages, when the cells have increased in 
number, various writers including Robert himself have been 
inclined to argue that the physical phenomena die away, and are 
overpowered and cancelled by agencies of a very different order. 
Here we pass into a region where direct observation and experi- 
ment are not at hand to guide us, and where a man’s trend of 
thought, and way of judging the whole evidence in the case, must 
shape his philosophy. We must remember that, even in a froth 
of soap-bubbles, we can apply an exact analysis only to the simplest 
cases and conditions of the phenomenon; we cannot describe, 
but can only imagine, the forces which in such a froth control the 
respective sizes, positions and curvatures of the innumerable 
bubbles and films of which it consists; but our knowledge is 
enough to leave us assured that what we have learned by in- 
vestigation of the simplest cases includes the principles which 
determine the most complex. In the case of the growing embryo 
we know from the beginning that surface tension is only one of 
the physical forces at work; and that other forces, including 
those displayed within the interior of each living cell, play their 
part in the determination of the system. But we have no evidence 
whatsoever that at this point, or that point, or at any, the dominion 
of the physical forces over the material system gives place to a 
new condition where agencies at present unknown to the physicist 
impose themselves on the living matter, and become responsible 
for the conformation of its material fabric. 


Before we leave for the present the subject of the segmenting 


344 | THE FORMS OF TISSUES [CH. 


egg, we must take brief note of two associated problems: viz. 
(1) the formation and enlargement of the segmentation cavity, or 
central interspace around which the cells tend to group themselves 
in a single layer, and (2) the formation of the gastrula, that is to 
say (in a typical case) the conversion “by invagination,” of the — 
one-layered ball into a two-layered cup. Neither problem is free 
from difficulty, and all we can do meanwhile is to state them in 
general terms, introducing some more or less plausible assumptions. 

The former problem is comparatively easy, as regards the 
tendency of a segmentation cavity to enlarge, when once it has 
been established. We may then assume that subdivision of the 
cells is due to the appearance of a new-formed septum within each 
cell, that this septum has a tendency to shrink under surface 
tension, and that these changes will be accompanied on the whole 
by a diminution of surface energy in the system. This being so, 
it may be shewn that the volume of the divided cells must be less 
than it was prior to division, or in other words that part of their 
contents must exude during the process of segmentation*. 
Accordingly, the case where the segmentation cavity enlarges and 
the embryo developes into a hollow blastosphere may, under the 
circumstances, be simply described as the case where that outflow 
or exudation from the cells of the blastoderm is directed on the 
whole inwards. 

The physical forces involved in the invagination of the cell- 
layer to form the gastrula have been repeatedly discussed}, but 
the true explanation seems as yet to be by no means clear. The 
case, however, is probably not a very difficult one, provided that 
we may assume a difference of osmotic pressure at the two poles 
of the blastosphere, that is to say between the cells which are . 
being differentiated into outer and inner, into epiblast and hypo- 
blast. It is plain that a blastosphere, or hollow vesicle bounded 
by a layer of vesicles, is under very different physical conditions 
from a single, simple yesicle or bubble. The blastosphere has no 
effective surface tension of its own, such as to exert pressure on 


* Professor Peddie has given me this interesting and important result, but the 
mathematical reasoning is too lengthy to be set forth here. 

+ Cf. Rhumbler, Arch. f. Hntw. Mech. xtv, p. 401, 1902; Assheton, ibid. xxxt, 
pp. 46-78, 1910. 


vit] OR CELL-AGGREGATES 345 


its contents or bring the whole into a spherical form; nor will local 
variations of surface energy be directly capable of affecting the 
form of the system. But if the substance of our blastosphere be 
sufficiently viscous, then osmotic forces may set up currents 
which, reacting on the external fluid pressure, may easily cause 
modifications of shape; and the particular case of invagination 
itself will not be difficult to account for on this assumption of 
non-uniform exudation and imbibition. 


CHAPTER VIII 
THE FORMS OF TISSUES OR CELL-AGGREGATES (continued) 


The problems which we have been considering, and especially 
that of the bee’s cell, belong to a class of “isoperimetrical” 
problems, which deal with figures whose surface is a minimum for 
a definite content or volume. Such problems soon become 
difficult, but we may find many easy examples which lead us 
towards the explanation of biological phenomena; and the 
particular subject which we shall find most easy of approach is 
that of the division, in definite proportions, of some definite 
portion of space, by a partition-wall of minimal area. The 
theoretical principles so arrived at we shall then attempt to apply, 
after the manner of Berthold and Errera, to the actual biological 
phenomena of cell-division. 

This investigation we may approach in two ways: by con- 
sidering, namely, the partitioning off from some given space or 
area of one-half (or some other fraction) of its content; or again, 
by dealing simultaneously with the partitions necessary for the 
. breaking up of a given space into a definite number of compart- 
ments. 

If we take, to begin with, the simple case of a cubical cell, it 
is obvious that, to divide it into two halves, the smallest possible 
partition-wall is one which runs parallel to, and midway between, 
two of its opposite sides. If we call a the length of one of the 
edges of the cube, then a? is the area, alike of one of its sides, and 
of the partition which we have interposed parallel, or normal, 
thereto. But if we now consider the bisected cube, and wish to 
divide the one-half of it again, it is obvious that another partition 
parallel to the first, so far from being the smallest possible, is 
precisely twice the size of a cross-partition perpendicular to it; 


CH. VII] OF SACHS’S RULES 347 


for the area of this new partition is a x a/2. And again, for a 
third bisection, our next partition must be perpendicular to the 
other two, and it is obviously a little square, with an area of 
(2a)? = 7a? 

From this we may draw the simple rule that, for a rectangular 
body or parallelopiped to be divided equally by means of a 
partition of minimal area, (1) the partition must cut across the 
longest axis of the figure; and (2) in the event of successive 
bisections, each partition must run at right angles to its immediate 
predecessor. 


Fig. 136. (After Berthold.) 


b) 


We have already spoken of “Sachs’s Rules,” which are an 
empirical statement of the method of cell-division in plant-tissues ; 
and we may now set them forth in full. 

(1) The cell typically tends to divide into two co-equal parts. 

(2) Each new plane of division tends to intersect at right 
angles the preceding plane of division. 

The first of these rules is a statement of physiological fact, 
not without its exceptions, but so generally true that it will 
justify us in lhmiting our enquiry, for the most part, to cases of 
equal subdivision. That it is by no means universally true for 
cells generally is shewn, for instance, by such well-known cases 


348 THE FORMS OF TISSUES [cH. 


as the unequal segmentation of the frog’s egg. It is true when the 
dividing cell is homogeneous, and under the influence of symmetrical 
forces; but it ceases to be true when the field is no longer dynami- 
cally symmetrical, for instance, when the parts differ in surface 
tension or internal pressure. This latter condition, of asymmetry 
of field, is frequent in segmenting eggs*, and is then equivalent 
to the principle upon which Balfour laid stress, as leading to 
“unequal” or to “partial” segmentation of the egg,—viz. the 
unequal or asymmetrical distribution of protoplasm and of food- 
yolk. 

The second rule, which also has its exceptions, is true in a 
large number of cases; and it owes its validity, as we may judge 
from the illustration of the repeatedly bisected cube, solely to the 
guiding principle of minimal areas. It is in short subordinate 
to, and covers certain cases included under, a much more important 
and fundamental rule, due not to Sachs but to Errera; that (3) the 
incipient partition-wall of a dividing cell tends to be such that its 
area is the least possible by which the given space-content can be 
enclosed. 


Let us return to the case of our cube, and let us suppose that, 
instead of bisecting it, we desire to shut off some small portion 
only of its volume. It is found in the course of experiments upon 
soap-films, that if we try to bring a partition-film too near to one 
side of a cubical (or rectangular) space, it becomes unstable; and 
is easily shifted to a totally new position, in which it constitutes 
a curved cylindrical wall, cutting off one corner of the cube. 
It meets the sides of the cube at right angles (for reasons which we 
have already considered); and, as we may see from the symmetry 


* M. Robert (J. c. p. 305) has compiled a long list of cases among the molluscs 
and the worms, where the initial segmentation of the egg proceeds by equal or 
unequal division. The two cases are about equally numerous. But like many 
other writers, he would ascribe this equality or inequality rather to a provision 
for the future than to a direct effect of immediate physical causation: “Ii semble 
assez probable, comme on l’a dit souvent, que la plus grande taille d’un blastomére 
est liée & importance et au développement précoce des parties du corps qui doivent 
en naitre: il y aurait la une sorte de reflet des stades postérieures du développement 
sur les premieres phénoménes, ce que M. Ray Lankester appelle precocious segrega- 
tion. I) faut avouer pourtant qu’on est parfois assez embarrassé pour assigner une 
cause a pareilles différences.”’ 


vit] OF AREAE MINIMAE 349 


of the case, it constitutes precisely one-quarter of a cylinder. 
Our plane transverse partition, wherever it was placed, had always 
the same area, viz. a7; and it is obvious that a cylindrical wall, 
if it cut off a small corner, may be much less than this. We want, 
accordingly, to determine what is the particular volume which 
might be partitioned off with equal economy of wall-space in one 
way as the other, that is to say, what area of cylindrical wall 
would be neither more nor less than the area a?._ The calculation 
is very easy. 

The swrface-area of a cylinder of length a is 27r.a, and that 
of our quarter-cylinder is, therefore, a. wr/2; and this being, by 
hypothesis, = a”, we have a= ar/2, or r = 2a/z. 

The volume of a cylinder, of length a, is avr?, and that of our 
quarter-cylinder is a . wr?/4, which (by substituting the value of 7) 
is equal to a3/z. 

Now precisely this same volume is, obviously, shut off by a 
transverse partition of area a?, if the third side of the rectangular 
space be equal to a/z. And this fraction, if we take a= 1, is 
equal to 0-318..., or rather less than one-third. And, as we have 
just seen, the radius, or side, of the corresponding quarter-cylinder 
will be twice that fraction, or equal to -636 times the side of the 
cubical cell. | 

If then, in the process of division 
of a cubical cell, it so divide that the 
two portions be not equal in volume 
but that one portion by anything less 
than about three-tenths of the whole, 
or three-sevenths of the other portion, 
there will be a tendency for the cell 
to divide, not by means of a plane 
transverse partition, but by means of 
a curved, cylindrical wall cutting off 
one corner of the original cell; and 
the part so cut off will be one-quarter of a cylinder. 

By a similar calculation we can shew that a spherical wall, 
cutting off one solid angle of the cube, and constituting an octant 
of a sphere, would likewise be of less area than a plane partition 
as soon as the volume to be enclosed was not greater than about 


Fig. 137. 


350 THE FORMS OF TISSUES [cH. 


one-quarter of the original cell*. But while both the cylindrical 
wall and the spherical wall would be of less area than the plane 
transverse partition after that limit (of one-quarter volume) was 
passed, the cylindrical would still be the better of the two up to 
a further limit. It is only when the volume to be partitioned off 


Fraction of cube partitioned off, 


partitions 


transverse partition 


3 
iy 
o 
Xx 
vu 
my 
v 
$s 
wih 
iS) 
v 
.y) 
< 
SS 


Fig. 138. 


* The principle is well illustrated in an experiment of Sir David Brewster’s 
(Trans. R.S.E. xxv, p. 111, 1869). A soap-film is drawn over the rim of a wine- 
glass, and then covered by a watch-glass. The film is inclined or shaken till it 
becomes attached to the glass covering, and it then immediately changes place, 
leaving its transverse position to take up that of a spherical segment extending 
from one side of the wine-glass to its cover, and so enclosing the same volume of 
air as formerly but with a great economy of surface, precisely as in the case of our 
spherical partition cutting off one corner of a cube. 


VIII | OF AREAE MINIMAE 351 


is no greater than about 0-15, or somewhere about one-seventh, 
of the whole, that the spherical cell-wall in an angle of the cubical 
cell, that is to say the octant of a sphere, is definitely of less area 
than the quarter-cylinder. In the accompanying diagram (Fig. 138) 
the relative areas of the three partitions are shewn for all fractions, 
less than one-half, of the divided cell. 


In this figure, we see that the plane transverse partition, whatever fraction 
of the cube it cut off, is always of the same dimensions, that is to say is 
always equal to a?, or = 1. If one-half of the cube have to be cut off, this 
plane transverse partition is much the best, for we see by the diagram that a 
cylindrical partition cutting off an equal volume would have an area about 
25%, and a spherical partition would have an area about 50% greater. 
The point A in the diagram corresponds to the point where the cylindrical 
partition would begin to have an advantage over the plane, that is to say 
(as we have seen) when the fraction to be cut off is about one-third, or -318 
of the whole. In like manner, at B the spherical octant begins to have an 
advantage over the plane; and it is not till we reach the point C that the 
spherical octant becomes of less area than the quarter-cylinder. 


The case we have dealt with is of little practical importance to 
the biologist, because the cases in which 
a cubical, or rectangular, cell divides 
unequally, and unsymmetrically, are 
apparently few; but we can find, as 
Berthold poimted out, a few examples, 
for instance in the hairs within the 
reproductive “conceptacles” of certain 
Fuci (Sphacelaria, etc., Fig. 139), or in 
the “paraphyses” of mosses (Fig. 142). 
But it is of great theoretical importance: as serving to introduce 
us to a large class of cases, in which the shape and the relative 
dimensions of the original cavity lead, according to the principle 
of minimal areas, to cell-division in very definite and sometimes 
unexpected ways. It is not easy, nor indeed possible, to give a 
generalised account of these cases, for the limiting conditions 
are somewhat complex, and the mathematical treatment soon 
becomes difficult. But it is easy to comprehend a few simple 
cases, which of themselves will carry us a good long way; and 
which will go far to convince the student that, in other cases 


A B 
Fig. 139. 


352 THE FORMS OF TISSUES [CH. 


which we cannot fully master, the same guiding principle is at 
the root of the matter. 


The bisection of a solid (or the subdivision of its volume in 
other definite proportions) soon leads us into a geometry which, 
if not necessarily difficult, is apt to be unfamiliar; but in such 
problems we can go a long way, and often far enough for our 
particular purpose, if we merely consider the plane geometry of 
a side or section of our figure. For instance, in the case of the 
cube which we have been just considering, and in the case of the 
plane and cylindrical partitions by which it has been divided, it 
is obvious that, since these two partitions extend symmetrically: 
from top to bottom of our cube, that we need only consider (so 
far as they are concerned) the manner in which they subdivide 
the base of the cube. The whole problem of the solid, up to a 
certain point, is contained in our plane diagram of Fig. 138. And 
when our particular solid is a solid of revolution, then it is obvious — 
that a study of its plane of symmetry (that is to say any plane 
passing through its axis of rotation) gives us the solution of the 
whole problem. The right cone is a case in point, for here the 
investigation of its modes of symmetrical subdivision is completely 
met by an examination of the isosceles triangle which constitutes 
its plane of symmetry. 

The bisection of an isosceles triangle by a line which shall 
be the shortest possible is a very easy problem. Let ABC be 
such a triangle of which A is the apex; it may be shewn that, 
for its shortest line of bisection, we are limited to three cases: 
viz. to a vertical line AD, bisecting the angle at A and the side 
BC; to a transverse line parallel to the base BC; or to an oblique 
line parallel to AB or to AC. The respective magnitudes, or 
lengths, of these partition lines follow at once from the magnitudes 
of the angles of our triangle. For we know, to begin with, since 
the areas of similar figures vary as the squares of their linear 
dimensions, that, in order to bisect the area, a line parallel to one 
side of our triangle must always have a length equal to 1/\/2 
of that side. If then, we take our base, BC, in all cases of 
a length = 2, the transverse partition drawn parallel to it will 
always have a length equal to 2/,/2, or =+/2. The vertical 


vit] OF AREAE MINIMAE 353 


partition, AD, since BD = 1, will always equal tanf (f being 
the angle ABC). And the oblique partition, GH, being equal to 
- AB/\/2=1/V2cosB. If then we call our vertical, transverse 


Fig. 140. 


and oblique partitions, V, 7, and O, we have V = tan; 
T=/2; and O= 1/2 cos B, or 


Viel: 0 = tan 8/4/2- 1.2 1/2 cos B. 


And, working out these equations for various values of B, we 
very soon see that the vertical partition (V) is the least of the 
three until 6 = 45°, at which limit V and O are each equal to 
1/\/2 = -707; and that again, when B = 60°, O and T are each 
= 1, after which 7 (whose value always = 1) is the shortest of 
the three partitions. And, as we have seen, these results are at 
once applicable, not only to the case of the plane triangle, but 
also to that of the conical cell. 


Fig. 141. 


In lke manner, if we have a spheroidal body, less than 
a hemisphere, such for instance as a low, watch-glass shaped 
cell (Fig. 141, a), it is obvious that the smallest possible 
partition by which we can divide it into two equal halves 


ToaGe 23 


354 THE FORMS OF TISSUES [CH. 


is (as in our flattened disc) a median vertical one. And 
likewise, the hemisphere itself can be bisected by no smaller 
partition meeting the walls at right angles than that median 
one which divides it into two similar quadrants of a sphere. 
But if we produce our hemisphere into a more elevated, conical 
body, or into a cylinder with spherical cap, it is obvious that there 
comes a point where a transverse, horizontal partition will bisect 
the figure with less area of partition-wall than a median vertical 
one (c). And furthermore, there will be an intermediate region, 
a region where height and base have their relative dimensions 
nearly equal (as in 6), where an oblique partition will be better 
than either the vertical or the transverse, though here the analogy 
of our triangle does not suffice to. give us the precise limiting 
values. We need not examine these limitations in detail, but we 
must look at the curvatures which accompany the several con- 
ditions. We have seen that a film tends to set itself at equal 
angles to the surface which it meets, and therefore, when that 
surface is a solid, to meet it (or its tangent if it be a curved surface) 
at right angles. Our vertical partition is, therefore, everywhere 
normal to the original cell-walls, and constitutes a plane surface. 

But in the taller, conical cell with transverse partition, the 
latter still meets the opposite sides of the cell at right angles, and 
it follows that it must itself be curved; moreover, since the 
tension, and therefore the curvature, of the partition is every- 
where uniform, it follows that its curved surface must be a portion 
of a sphere, concave towards the apex of the original, now divided, 
cell. In the intermediate case, where we have an oblique partition, 
meeting both the base and the curved sides of the mother-cell, 
the contact must still be everywhere at right angles: provided 
we continue to suppose that the walls of the mother-cell (like those 
of our diagrammatic cube) have become practically rigid before 
the partition appears, and are therefore not affected and deformed 
by the tension of the latter. In such a case, and especially when 
the cell is elliptical in cross-section, or is still more complicated 
in form, it is evident that the partition, in adapting itself to 
circumstances and in maintaining itself as a surface of minimal 
area subject to all the conditions of the case, may have to assume 
a complex curvature. 


vit] OF SIGMOID OR S-SHAPED PARTITIONS 355 


While in very many cases the partitions (like the walls of the 
original cell) will be either plane or spherical, a more complex 
curvature will be assumed under a variety of conditions. It will 
be apt to occur, for instance, when the mother-cell is irregular in 
shape, and one particular case of such asymmetry will be that in 
which (as in Fig. 143) the cell has begun to braich, or give off a 
diverticulum, before division takes place. A very complicated 
case of a different kind, though not without its analogies to the 
cases we are considering, will occur in the partitions of minimal 
area which subdivide the spiral tube of a nautilus, as we shall 


aa taae 


HS HI 


Fig. 142. ae partitions: A, from Taonia atomaria (after Reinke); B, from 
paraphyses of Fucus; C, from rhizoids of Moss; D, from paraphyses of 
Polytrichum. 

presently see. And again, whenever we have a marked internal 
asymmetry of the cell, leading to irregular and anomalous modes 
of division, in which the cell is not necessarily divided into two 
equal halves and in which the partition-wall may assume an 
oblique position, then apparently anomalous curvatures will tend 
to make their appearance*. 

Suppose that a more or less oblong cell have a tendency to 
divide by means of an oblique partition (as may happen through 
various causes or conditions of asymmetry), such a partition will 
still have a tendency to set itself at right angles to the rigid walls 


* Cf. Wildeman, Attache des Clo‘sons, ete., pls. 1. 2. 
23—2 


a 


356 THE FORMS OF TISSUES [cH. 


of the mother-cell: and it will at once follow that our oblique 
partition, throughout its whole extent, will assume the form of 
a complex, saddle-shaped or anticlastic surface. 

Many such cases of partitions with complex or double curvature 
exist, but they are not always easy of recognition, nor is the 
particular case where they appear in a terminal cell a common 
one. We may see them, for instance, in the roots (or rhizoids) 
of Mosses, especially at the point of development of a new rootlet 
(Fig. 142, C); and again among Mosses, in the “paraphyses” of 
the male prothalli (e.g. in Polytrichum), we firid more or less 
similar partitions (D). They are frequent also among many Fuci, 
as in the hairs or paraphyses of Fucus itself (B). In Taonia 


A B (e D 


Fig. 143. Diagrammatic explanation of S-shaped partition. 


atomaria, as figured in Reinke’s memoir on the Dictyotaceae of. 
the Gulf of Naples*, we see, in like manner, oblique partitions, 
which on more careful examination are seen to be curves of 
double curvature (Fig. 142, A). 
The physical cause and origin of these S-shaped partitions is 
somewhat obscure, but we may attempt a tentative explanation. 
When we assert a tendency for the cell to divide transversely to 
its long axis, we are not only stating empirically that the partition 
tends to appear in a small, rather than a large cross-section of the 
cell: but we are also implicitly ascribing to the cell a longitudinal 
polarity (Fig. 143, A), and implicitly asserting that it tends to 


* Nova Acta K. Leop. Akad. x1, 1, pl. Iv. 


/ 


vit] |OF SIGMOID OR 8-SHAPED PARTITIONS 357 


divide (just as the segmenting egg does), by a partition transverse 
to its polar axis. Such a polarity may conceivably be due to 
a chemical asymmetry, or anisotropy, such as we have learned 
of (from Professor Macallum’s experiments) in our chapter on 
Adsorption. Now if the chemical concentration, on which this 
anisotropy or polarity (by hypothesis) depends, be unsymmetrical, 
one of its poles being as it were deflected to one side, where a little 
branch or bud is being (or about to be) given off,—all in precise 
accordance with the adsorption phenomena described on p. 289,— 
then our “polar axis’’ would necessarily be a curved axis, and the 
partition, being constrained (again ex hypothesz) to arise transversely 
to the polar axis, would lie obliquely to the apparent axis of the 
cell (Fig. 143, B, C). And if the oblique partition be so situated 
that it has to meet the opposite walls (as in C), then, in order to 
do so symmetrically (i.e. either perpendicularly, as when the 
cell-wall is already solidified, or at least at equal angles on either 
side), it is evident that the partition, in its course from one side 
of the cell to the other, must necessarily assume a more or less 
S-shaped curvature (Fig. 143, D). 

As a matter of fact, while we have abundant simple illustrations 
of the principles which we have now begun to study, apparent 
exceptions to this simplicity, due to an asymmetry of the cell 
itself, or of the system of which the single cell is but a part, are 
by no means rare. For example, we know that in cambium-cells, 
division frequently takes place parallel to the long axis of the 
cell, when a partition of much less area would suffice if it were 
set cross-ways: and it is only when a considerable disproportion 
has been set up between the length and breadth of the cell, that 
the balance is in part redressed by the appearance of a transverse 
partition. It was owing to such exceptions that Berthold was 
led to qualify and even to depreciate the importance of the law 
of minimal areas as a factor in cell-division, after he himself had 
done so much to demonstrate and elucidate it*. He was deeply 
and rightly impressed by the fact that other forces besides surface 

* Cf. Protoplasmamechanik, p. 229: “Insofern liegen also die Verhaltnisse hier 
wesentlich anders als bei der Zertheilung hohler Kérperformen durch fliissige 
Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der 


kleinsten Flichen geforderte Lage und Kriimmung annimmt, so werden wir den 
Grund dafiir in andrer Weise abzuleiten haben.” 


358 THE FORMS OF TISSUES [CH. 


tension, both external and internal to the cell, play their part 
in the determination of its partitions, and that the answer to 
our problem is not to be given in a word. How fundamentally 
important it is, however, in spite of all conflicting tendencies and 
apparent exceptions, we shall see better and better as we proceed. 


But let us leave the exceptions and return to a consideration 
of the simpler and more general phenomena. And in so doing, 
let us leave the case of the cubical, quadrangular or cylindrical 
cell, and examine the case of a spherical cell and of its successive 
divisions, or the still simpler case of a circular, discoidal cell. 

When we attempt to investigate mathematically the position 
and form of a partition of minimal area, it is plain that we shall 
be dealing with comparatively simple cases wherever even one | 
dimension of the cell is much less than the other two. Where two 
dimensions are small compared with the third, as in a thin cylin- 
drical filament hke that of Spirogyra, we have the problem at its 
simplest; for it is at once obvious, then, that the partition must 
lie transversely to the long axis of the thread. But even where 
one dimension only is relatively small, as for instance in a flattened » 
plate, our problem is so far simplified that we see at once that the 
partition cannot be parallel to the extended plane, but must cut 
the cell, somehow, at right angles to that plane. In short, the 
problem of dividing a much flattened solid becomes identical with 
that of dividing a simple surface of the same form. 

There are a number of small Algae, growing in the form of 
small flattened discs, consisting (for a time at any rate) of but a 
single layer of cells, which, as Berthold shewed, exemplify this 
comparatively simple problem; and we shall find presently that 
it is also admirably illustrated in the cell-divisions which occur in 
the egg of a frog or a sea-urchin, when the egg for the sake of 
experiment is flattened out under artificial pressure. 

Fig. 144 (taken from Berthold’s Monograph of the Naples 
Bangiaciae) represents younger and older discs of the little alga 
Erythrotrichia discigera; and it will be seen that, in all stages save 
the first, we have an arrangement of cell-partitions which looks 
somewhat complex, but into which we must attempt to throw some 
light and order. Starting with the original single, and flattened, 


vur} THE SEGMENTATION OF A DISC 359 


cell, we have no difficulty with the first two cell-divisions; for 
we know that no bisecting partitions can possibly be shorter than 
the two diameters, which divide the cell into halves and into 


SEAMEN 


ie 


Fig. 144. Development of Erythrotrichia. (After Berthold.) 


quarters. We have only to remember that, for the sum total of 
partitions to be a minimum, three only must meet in a point; 
and therefore, the four quadrantal walls must shift a little, pro- 
- ducing the usual little median partition, or cross-furrow, instead 
of one common, central point of junction. This little inter- 
mediate wall, however, will be very small, and to all intents and 
purposes we may deal with the 
case as though we had now to do 
with four equal cells, each one of 
them a perfect quadrant. And 
so our problem is, to find the 
shortest line which shall divide the 
quadrant of a circle into two 
halves of equal area... A radial 
partition (Fig. 145, a), starting 
from the apex of the quadrant, is 
at once excluded, for a reason 
similar to that just referred to; 
our choice must lie therefore between two modes of division such 
- as are illustrated in Fig. 145, where the partition is either (as in B) 


Fig. 145. 


360 THE FORMS OF TISSUES [cH. 


concentric with the outer border of the cell, or else (as in c) cuts 
that outer border; in other words, our partition may (B) cut both 
radial walls, or (c) may cut one radial wall and the periphery. 
These are the two methods of division which Sachs called, respec- 
tively, (B) periclinal, and (c) anticlinal*. We may either treat the 
walls of the dividing quadrant as already solidified, or at least as 
having a tension compared with which that of the incipient 
partition film is inconsiderable. In either case the partition must 
meet the cell-wall, on either side, at right angles, and (its own 
tension and curvature being everywhere uniform) it must take the 
form of a circular are. 

Now we find that a flattened cell which is approximately a 
quadrant of a circle invariably divides after the manner of 
Fig. 145, c, that is to say, by an approximately circular, anticlinal 
wall, such as we now recognise in the eight-celled stage of 
Erythrotrichia (Fig. 144); let us then consider that Nature has 
solved our problem for us, and let us work out the actual 
geometric conditions. 

Let the quadrant OAB {in Fig. 146) be divided into two 
parts of equal area, by the circular are MP. It is required to 
determine (1) the position of P upon the are of the quadrant, 
that is to say the angle BOP; (2) the position of the point M 
on the side OA; and (3) the length of the arc MP in terms of a 
radius of the quadrant. 

(1) Draw OP; also PC a tangent, meeting OA in C; and 
PN, perpendicular to OA. Let us call aa radius; and @ the angle 
at C, which is obviously equal to OPN, or POB. Then 


CP =a cot@; PN—«acos 0; NC = CP cos? —@ - cos G/siaie. 
"The area of the portion PMN 

= Oreo +PN,. NG 

= ka? cot? 6 — 4a cos 0. a cos? 6/sin 6 


= 4a? (cot? 0 — cos? 6/sin 6). 


* There is, I think, some ambiguity or disagreement among botanists as to the 
use of this latter term: the sense in which I am using it, viz. for any partition 
which meets the outer or peripheral wall at right angles (the strictly radial partition 
being for the present excluded), is, however, clear. 


_ vo] THE BISECTION OF A QUADRANT 361 


And the area of the portion PNA 
= 4a? (7/2 — 0) —40N .NP 
= ha? (7/2 — 0) — fasin 0.acos@ 
= 4a? (7/2 — 0 — sin @. cos @). 
Therefore the area of the whole portion PMA 
= a?/2 (7/2 — 0+ 86 cot? 6 — cos? 8/sin 6 — sin 6 . cos 6) 
= a?/2 (7/2 — 0+ 6 cot? 6 — cot 8), 
and also, by hypothesis, = }. area of the quadrant, = za?/8. 
B 


O M N A Cc 
Fig. 146. 


Hence @ is defined by the equation 
a?/2 (7/2 — 6 + 0 cot? 6 — cot 0) = ma?/8, 
or 7/4 — 0+ 6 cot? @— cot 0= 0. 
We may solve this equation by constructing a table (of which 
the following is a small portion) for various values of @. 
6 1/4 -6 —cot@ +é4cot?@ =i 
34° 34’ ‘7854 — -6033 — 1:-4514 + 1-2709= -0016 
35’ °7854 = =-6036 =—-1-4505 1-2700 “0013 
36’ ‘7854 -6039 11-4496 1-2690 0009 
37’ . -7854 -6042 1-4487 1-2680 ‘0005 
38’ *7854 =-6045~—- 11-4478 1-2671 “0002 
39’ ‘7854 = -6048 = 11-4469 1-2661 — -0002 
40’ °7854 = 6051 ~—- 11-4460 1-2652 — -0005 


362 THE FORMS OF TISSUES . | CH. 


We see accordingly that the equation is solved (as accurately 
as need be) when @ is an angle somewhat over 34° 38’, or say 
34° 381’. That is to say, a quadrant of a circle is bisected by a 
circular arc cutting the side and the periphery of the quadrant 
at right angles, when the arc is such as to include (90° — 34° 38’), 
i.e. 55° 22’ of the quadrantal arc. 

This determination of ours is practically identical with that 
which Berthold arrived at by a rough and ready method, without 
the use of mathematics. He simply tried various ways of dividing 
a quadrant of paper by means of a circular arc, and went on doing 
so till he got the weights of his two pieces of paper approximately 
equal. The angle, as he thus determined it, was 34-6°, or say 
34° 36’. 

(2) The position of M on the side of the quadrant OA is 
given by the equation OM = acosec @—acot@; the value of 
which expression, for the angle which we have just discovered, 
is -3028. That is to say, the radius (or side) of the quadrant will 
be divided by the new partition into two parts, in the proportions 
of nearly three to seven. 

(3) The length of the arc MP is equal to a@ cot 6; and the 
value of this for the given angle is -8751. This is as much as to 
say that the curved partition-wall which we are considering is 
shorter than a radial partition in the proportion of 8? to 10, or 
seven-eights almost exactly. 

But we must also compare the length of this curved “ antielinalt 
partition-wall (MP) with that of the con- 
centric, or periclinal, one (RS, Fig. 147) by 
which the quadrant might also be bisected. 
The length of this partition is obviously 
equal to the arc of the quadrant (i.e. the 
peripheral wall of the cell) divided by 1/2;- 

O ™M A or, in terms of the radius, = 7/2./2 = 1-111. 

a Pie So that, not only is the anticlinal partition 

(such as we actually find in nature) notably the best, but the 

periclinal one, when it comes to dividing an entire quadrant, is 

very considerably larger even than a radial partition. 

The two cells into which our original quadrant is now divided, 
while they are equal in volume, are of very different shapes; the 


vit] THE BISECTION OF A QUADRANT 363 


one is a triangle (MAP) with two sides formed of circular arcs, 
and the other is a four-sided figure (WOBP), which we may call 
approximately oblong. We cannot say as yet how the triangular 
portion ought to divide; but it is obvious that the least possible 
partition-wall which shall bisect the other must run across the 
long axis of the oblong, that is to say periclinally. This, also, 1s 
precisely what tends actually to take place. In the following 
diagrams (Fig. 148) of a frog’s egg dividing under pressure, that 
is to say when reduced to the form of a flattened plate, we see, 
firstly, the division into four quadrants (by the partitions 1, 2); 
secondly, the division of each quadrant by means of an anti- 
clinal circular arc (3, 3), cutting the peripheral wall of the quadrant 
approximately in the proportions of three to seven; and thirdly, 


Fig. 148. Segmentation of frog’s egg, under artificial compression. 
(After Roux.) 


we see that of the eight cells (four triangular and four oblong) 
into which the whole egg is now divided, the four which we have 
called oblong now proceed to divide by partitions transverse to 
their long axes, or roughly parallel to the periphery of the egg. 


The question how the other, or triangular, portion of the divided 
qudarant will next divide leads us to another well-defined problem, ; 
which is only a slight extension, making allowance for the circular 
ares, of that elementary problem of the triangle we have already 
considered. We know now that an entire quadrant must divide 
(so that its bisecting wall shall have the least possible area) by 
means of an anticlinal partition, but how about any smaller 
sectors of circles? It is obvious in the case of a small prismatic 


364 THE FORMS OF TISSUES [cH. 


sector, such as that shewn in Fig. 149, that a periclinal partition. 

is the smallest by which we can possibly bisect the cell; we want, 

accordingly, to know the limits below which the periclinal partition — 
is always the best, and above which the anticlinal arc, as in the 

case of the whole quadrant, has the advantage in regard to small- 

ness of surface area. 

This may be easily determined; for the preceding investigation 
is a perfectly general one, and the results hold good for sectors 
of any other arc, as well as for the quadrant, or are of 90°. That 
is to say, the length of the partition-wall MP is always determined 
by the angle 0, according to our equation MP = a6 cot @; and 
the angle @ has a definite relation to a, the angle of arc. 


OA-V 2 
Oa- | 


O B 


Fig. 149. 


Moreover, in the case of the periclinal boundary, RS (Fig. 147) 
(or ab, Fig. 149), we know that, if it bisect the cell, 


RS =a.a/y/2. 
Accordingly, the are RS will be just equal to the are MP when 
6 cot 6 = a/r/2. 
When @ cot 06 >a/,\/2, or MP > RS, 
then division will take place as in RS. 
When 6 cot 6<a/,/2, or MP < RS, 


then division will take place as in MP. 

In the accompanying diagram (Fig. 150), I have plotted the 
various magnitudes with which we are concerned, in order to 
exhibit the several limiting values. Here we see, in the first 
place, the curve marked a, which shews on the (left-hand) vertical 
scale the various possible magnitudes of that angle (viz. the angle 


vir] THE BISECTION OF A QUADRANT 365 


of arc of the whole sector which we wish to divide), and on the 
horizontal scale the corresponding values of 0, or the angle which 


Angle (@) determining the intersection of the partition-wall with the outer border 
of the cell. 


| T 2-0 


die 


610° 20° 30° 40 50° 60. 70 80. 90° 


Og 


9° 
80|- 


ae 
fo.) 


70|— * 1-4 


a 


8, 


Angle of are (a) of the prismatic sector, or cell, undergoing division. 
nN 
Do 


= | [eS 
Fig. 150. 


determines the point on the periphery where it is cut by the 
partition-wall, MP. Two limiting cases are to be noticed here: 
(1) at 90° (point A in diagram), because we are at present only 


Lengths of the several partitions, in terms of a radius (r=1). 


366 THE FORMS OF TISSUES [CH. 


dealing with arcs no greater than a quadrant; and (2), the point 
(B) where the angle 6 comes to equal the angle a, for after that 
point the construction becomes impossible, since an anticlinal 
bisecting partition-wall would be partly outside the cell. The only 
partition which, after the poimt, can possibly exist, is a periclinal 
one. This point, as our diagram shews us, occurs when the angles 
(a and 6) are each rather under 52°. 

Next I have plotted, on the same diagram, and in relation to 
the same scales of angles, the corresponding lengths of the two 
partitions, viz. RS and MP, their lengths being expressed (on 
the right-hand side of the diagram) in relation to the radius of 
the circle (a), that is to say the side wall, OA, of our cell. 

The limiting values here are (1), C, C’, where the angle of are 
is 90°, and where, as we have already seen, the two partition-walls 
have the relative magnitudes of MP: RS = 0-875: 1-111; (2) the 
point D, where RS equals unity, that is to say where the periclinal 
partition has the same length as a radial one; this occurs when 
a is rather under 82° (cf. the pomts D, D’); (3) the point #, where 
RS and MP intersect; that is to say the point at which the two 
partitions, periclinal and anticlinal, are of the same magnitude; 
this is the case, according to our diagram, when the angle of arc 
is just over 623°. We see from this, then, that what we have 
called an anticlinal partition, as MP, is only likely to occur in 
a triangular or prismatic cell whose angle of arc les between 
90° and 623°. In all narrower or more tapering cells, the periclinal 
partition will be of less area, and will therefore be more and more 
likely to occur. : 

The case (Ff) where the angle a is just 60° is of some interest. 
Here, owing to the curvature of the peripheral border, and the 
consequent fact that the peripheral angles are somewhat greater 
than the apical angle a, the perichnal partition has a very slight 
and almost imperceptible advantage over the anticlinal, the 
relative proportions being about as MP: RS = 0-73: 0-72° Butit 
the equilateral triangle be a plane spherical triangle, i.e. a plane 
triangle bounded by circular arcs, then we see that there is no 
longer any distinction at all between our two partitions; MP 
and RS are now identical. 

On the same diagram, I have inserted the curve for values of 


vu] THE SEGMENTATION OF A DISC 367 


cosec 6 — cot d = OM, that is to say the distances from the centre, 
along the side of the cell, of the starting-point (1) of the anticlinal 
partition. The point C” represents its position in the case of 
a quadrant, and shews it to be (as we have already said) about 
3/10 of the length of the radius from the centre. If, on the other 
hand, our cell be an equilateral triangle, then we have to read off 
the point on this curve corresponding to a = 60°, and we find it 
at the point F’” (vertically under F), which tells us that the 
partition now starts 4-5/10, or nearly halfway, along the radial 
wall. 


The foregoing considerations carry us a long way in our 
investigations of many of the simpler forms of cell-division. 
Strictly speaking they are limited to the case of flattened cells, 
in which we can treat the problem as though we were simply 
partitioning a plane surface. But it is obvious that, though they 
do not teach us the whole conformation of the partition which 
divides a more complicated solid into two halves, yet they do, even 
in such a case, enlighten us so far, that they tell us the appearance 
presented in one plane of the actual solid. And as this is all that 
we see in a microscopic section, it follows that the results we have 
arrived at will greatly help us in the interpretation of microscopic 
appearances, even in comparatively complex cases of cell-division. 

Let us now return to our 
quadrant cell (OAPB), which we B' 
have found to be divided into 
a triangular and a quadrilateral 
portion, as in Fig. 147 or Fig. 151; 
and let us now suppose the whole 
_ system to grow, in a uniform 
fashion, as a prelude to further 
subdivision. The whole quadrant, 
growing uniformly (or with equal 
radial increments), will still re- 
main a quadrant, and it is a cual 
obvious, therefore, that for every 
new increment of size, more will 
be added to the margin of its triangular portion than to the 

‘ 


Fig. 151. 


368 THE FORMS OF TISSUES [cH. 


narrower margin of its quadrilateral portion; and these incre- 
ments will be in proportion to the angles of arc, viz. 55° 22’: 34° 38’, 
or as -96:-60, 1.e. as 8:5. And accordingly, if we may assume 
(and the assumption is a very plausible one), that, just as the 
quadrant itself divided into two halves after it got to a certain 
size, so each of its two halves will reach the same size before 
again dividing, it is obvious that the triangular portion will be 
doubled in size, and therefore ready to divide, a considerable 
time before the quadrilateral part. To work out the problem in 
detail would lead us into troublesome mathematics; but if 
we simply assume that the increments are proportional to the 
increasing radii of the circle, we have the following equations :— 

Let us call the triangular cell 7, and the quadrilateral, Q@ 
(Fig. 151); let the radius, OA, of the original quadrantal cell 
=a=1; and let the increment which is required to add on a 
portion equal to 7 (such as PP’A’A) be called xz, and let that 
required, similarly, for the doubling of Q be called 2’. 

Then we see that the area of the original quadrant 


= T + @Q = 41a? = -78540?, 
while the area of T — aga ae 
The area of the enlarged sector, p'OA’, 
= (a+ x)? x (55° 22’) + 2 = -4831 (a + a)?, 
and the area OPA 
= a? x (55° 22’) + 2 = -4831a?. 
Therefore the area of the added portion, T’, 
= -4831 {(a + 2)? — a}. 
And this, by hypothesis, 
= T = :3927a?. 
We get, accordingly, since a = 1, 
x? + 2e = -3927/-4831 = -810, 
and, solving, 
a+1=V1-81 = 1-345, or x = 0-345. 
Working out z’ in the same way, we arrive at the approximate 
value, 2 + 1 = 1-517. 


vu] THE SEGMENTATION OF A DISC 369 


This is as much as to say that, supposing each cell tends to 
divide into two halves when (and not before) its original size is 
doubled, then, in our flattened disc, the triangular cell 7 will tend 
to divide when the radius of the disc has increased by about a 
third (from 1 to 1-345), but the quadrilateral cell, Y, will not tend 
to divide until the linear dimensions of the disc have increased 
by about a half (from | to 1-517). 

The case here illustrated is of no small general importance. 
For it shews us that a uniform and symmetrical growth of the 
organism (symmetrical, that is to say, under the limitations of a 
plane surface, or plane section) by no means involves a uniform 
or symmetrical growth of the individual cells, but may, under 
certain conditions, actually lead to inequality among these; and 
this inequality may be further emphasised by differences which 
arise out of it, in regard to the order of frequency of further 
‘subdivision. This phenomenon (or to be quite candid, this 
hypothesis, which is due to Berthold) is entirely independent of 
any change or variation in individual surface tensions; and 
accordingly it is essentially different from the phenomenon of 
unequal segmentation (as studied by Balfour), to which we have 
referred on p. 348. 

In this fashion, we might go on to consider the manner, and 
the order of succession, in which the subsequent cell-divisions 
would tend to take place, as governed by the principle of minimal 
areas. But the calculations would grow more difficult, or the 
results got by simple methods would grow less and less exact. 
At the same time, some of these results would be of great interest, 
_and well worth the trouble of obtaining. For instance, the precise 
manner in which our triangular cell, 7, would next divide would 
be interesting to know, and a general solution of this problem is 
certainly troublesome to calculate. But in this particular case 
we can see that the width of the triangular cell near P is so 
obviously less than that near either of the other two angles, that 
a circular are cutting off that angle is bound to be the shortest 
possible bisecting line; and that, in short, our triangular cell 
will tend to subdivide, just lke the original quadrant, into a 
triangular and a quadrilateral portion. 

But the case will be different next time, because in this new 


Tila (Obs 24 


370 THE FORMS OF TISSUES [CH. 


triangle, PRQ, the least width is near the innermost angle, that 
at Q; and the bisecting circular are will therefore be opposite to Q, 
or (approximately) parallel to PR. The importance of this fact is 
at once evident; for it means to say that there soon comes a 
time when, whether by the division of triangles or of quadrilaterals, 
we find only quadrilateral cells adjoiming the periphery of our 
circular disc. In the subsequent division of these quadrilaterals, 
the partitions will arise transversely to their long axes, that is to 
say, radially (as U, V); and we shall consequently have a super- 
ficial or peripheral layer of quadrilateral cells, with sides approxi- 
mately parallel, that is to say what we are accustomed to call an_ 
epidermis. And this epidermis or superficial layer will be’in clear 
contrast with the more irregularly shaped cells, the products of 
triangles and quadrilaterals, which make up the deeper, underlying 
layers of tissue. 


= : 
Pe 


P 
O M A 
Fig. 152. 

In following out these theoretic principles and others like to 
them, in the actual division of living cells, we must always bear 
in mind certain conditions and qualifications. In the first place, 
the law of minimal area and the other rules which we have arrived 
at are not absolute but relative: they are links, and very important 
links, in a chain of physical causation; they are always at work, 
but their effects may be overridden and concealed by the operation 
of other forces. Secondly, we must remember that, in the great 
majority of cases, the cell-system which we have in view is con- 
stantly increasing in magnitude by active growth; and by this 
means the form and also the proportions of the cells are continually 
liable to alteration, of which phenomenon we have already had 
an example. Thirdly, we must carefully remember that, until 
our cell-walls become absolutely solid and rigid, they are always 
apt to be modified in form owing to the tension of the adjacent 


vur} THE SEGMENTATION OF A DISC 371 


walls; and again, that so long as our partition films are fluid or 
semifluid, their points and lines of contact with one another may 
shift, like the shifting outlines of a system of soap-bubbles. This 
is the physical cause of the movements frequently seen among 
segmenting cells, like those to which Rauber called attention in 
‘ the segmenting ovum of the frog, and like those more striking 
movements or accommodations which give rise to a so-called 
“spiral” type of segmentation. 


Bearing in mind, then, these considerations, let us see what 
our flattened disc is likely to look like, after a few successive 


c d 


Fig. 153. Diagram of flattened or discoid cell dividing into octants: to shew 
gradual tendency towards a position of equilibrium. 


divisions into component cells. In Fig. 153, a, we have a diagram- 
matic representation of our disc, after it has divided into four 
quadrants, and each of these in turn into a triangular and a 
quadrilateral portion; but as yet, this figure scarcely suggests 
to us anything like the normal look of an aggregate of living cells. 
But let us go a little further, still limiting ourselves, however, 
to the consideration of the eight-celled stage. Wherever one of 
our radiating partitions meets the peripheral wall, there will (as 
we know) be a mutual tension between the three convergent films, 
which will tend to set their edges at equal angles to one another, 
angles that is to say of 120°. In consequence of this, the outer 
wall of each individual cell will (in this surface view of our disc) 
24-2 


372 THE FORMS OF TISSUES [CH. 


be an arc of a circle of which we can determine the centre by the 
method used on p. 307; and, furthermore, the narrower cells, 
that is to say the quadrilaterals, will have this outer border 
somewhat more curved than their broader neighbours. We arrive, 
then, at the condition shewn in Fig. 153, b. Within the cell, 
also, wherever wall meets wall, the angle of contact must tend, 
in every case, to be an angle of 120°; and in no case may more 
than three films (as seen in section) meet in a point (c); and 
this condition, of the partitions meeting three by three, and at 
co-equal angles, will obviously involve the curvature of some, if 
not all, of the partitions (d) which in our preliminary investigation 
we treated as plane. To solve this problem in a general way is 
no easy matter; but it 1s a problem which Nature solves in 
every case where, as in the case we are considering, eight bubbles, 
or eight cells, meet together in a (plane or curved) surface. An 
approximate solution has been given in Fig. 153, d; and it will now 
at once be recognised that this figure has vastly more resemblance 
to an aggregate of living cells than had the diagram of Fig. 153, a 
with which we began. 

Just as we have constructed in this case a series of purely 
diagrammatic or schematic figures, so it will be as a rule possible 
to diagrammatise, with but little alteration, the 
complicated appearances presented by any ordinary 
aggregate of cells. The accompanying little figure 
(Fig. 154), of a germinating spore of a Liverwort 
(Riccia), after a drawing of Professor Campbell’s, 
scarcely needs further explanation: for it is well-nigh a 
typical diagram of the method of space-partitioning which we are 
now considering. Let us look again at our figures (on p. 359) of the 
dise of Erythrotrichia, from Berthold’s Monograph of the Bangiaceae 
and redraw the earlier stages in diagrammatic fashion. In the 
following series of diagrams the new partitions, or those just about 
to form, are in each case outlined; and in the next succeeding 
stage they are shewn after settling down into position, and after 
exercising their respective tractions on the walls previously laid 
down. It is clear, I think, that these four diagrammatic figures 
represent all that is shewn in the first five stages drawn by 
Berthold from the plant itself; but the correspondence cannot 


Fig. 154. 


vur| THE SEGMENTATION OF A DISC 373 


in this case be precisely accurate, for the simple reason that 
Berthold’s figures are taken from different individuals, and are 
therefore only approximately consecutive and not strictly con- 
tinuous. The last of the six drawings in Fig. 144 is already too 


Fig. 155. Theoretical arrangement of successive partitions in a discoid 
cell; for comparison with Fig. 144. 


complicated for diagrammatisation, that is to say it is too com- 
plicated for us to decipher with certainty the precise order of 
appearance of the numerous partitions which it contains. But 
in Fig. 156 I shew one more diagrammatic figure, of a dise which 


Fig. 156. Theoretical.division of a discoid cell into sixty-four chambers: no 
allowance being made for the mutual tractions of the cell-walls. 


has divided, according to the theoretical plan, into about sixty- 
four cells; and making due allowance for the successive changes 
which the mutual tensions and tractions of the partitions must 


374 THE FORMS OF TISSUES [CH. 


bring about, increasing in complexity with each succeeding stage, 
we can see, even at this advanced and complicated stage, a very 
considerable resemblance between the actual picture (Fig. 144) 
and the diagram which we have here constructed in obedience to 
a few simple rules. 

In like manner, in the annexed figures, representing sections 
through a voung embryo of a Moss, we have very little difficulty 
in discerning the successive stages that must have intervened 
between the two stages shewn: so as to lead from the just divided 
quadrants (one of which, by the way, has not yet divided in our 
figure (a)) to the stage (b) in which a well-marked epidermal 
layer surrounds an at first sight irregular agglomeration of 
“fundamental”’ tissue. 


a b 
Fig. 157. Sections of embryo of a moss. (After Kienitz-Gerloff.) 


In the last paragraph but one, I have spoken of the difficulty 
of so arranging the meeting-places of a number of cells that at 
each junction only three cell-walls shall meet in a line, and all 
three shall meet it at equal angles of 120°. As a matter of fact, the 
problem is soluble in a number of ways; that is to say, when we 
have a number of cells, say eight as in the case considered, enclosed 
in a common boundary, there are various ways in which their 
walls can be made to meet internally, three by three, at equal 
angles; and these differences will entail differences also in the 
curvature of the walls, and consequently in the shape of the cells. 
The question is somewhat complex; it has been dealt with by 
Plateau, and treated mathematically by M. Van Rees*. 

If within our boundary we have three cells all meeting 


* Cit. Plateau, Statique-des Liquides, i, p. 358. 


vit] THE PARTITIONING OF SPACE 375 


internally, they must meet in a point; furthermore, they tend to 
do so at’ equal angles of 120°, and there is an end of the matter. 
If we have four cells, then, as we have already seen, the conditions 
are satisfied by interposing a little intermediate wall, the two 
extremities of which constitute the meeting-points of three cells 
each, and the upper edge of which marks the “polar furrow.” 
Similarly, in the case of five cells, we require two little intermediate 
walls, and two polar furrows; and we soon arrive at the rule that, 
for n cells, we require » — 3 little longitudinal partitions (and 
corresponding polar furrows), connecting the triple junctions of 


4 5 6 u 8 Cells ees 
Bigeye Nad where 
' i BNO 
ea en Nanety OU aor FG 
eee 
Te ee 
eel 
ie 
CEG. 


Fig. 158. Various possible arrangements of intermediate partitions, in 
groups of 4, 5, 6, 7 or 8 cells. 


the cells; and these little walls, like all the rest within the system, 
must be inclined to one another at angles of 120°. Where we 
have only one such wall (as in the case of four cells), or only two 
(as in the case of five cells), there is no room for ambiguity. But 
where we have three little connecting-walls, as in the case of six 
cells, it is obvious that we can arrange them in three different 
ways, as in the annexed Fig. 159. In the system of seven cells, 
the four partitions can be arranged in four ways; and the five 
partitions required in the case of eight cells can be arranged in no 
less than thirteen different ways, of which Fig. 158 shews some 
half-dozen only. It does not follow that, so to speak, these various 


376 THE FORMS OF TISSUES [CH. 


arrangements are all equally good; some are known to be much 
more stable than others, and some have never yet been realised 
in actual experiment. 

The conditions which lead to the presence of any one of them, 
in preference to another, are as yet, so far as I am aware, un- 
determined, but to this point we shall return. 


Examples of these various arrangements meet us at every 
turn, and not only in cell-aggregates, but in various cases where 
non-rigid and semi-fluid partitions (or partitions that were so to 
begin with) meet together. And it is a necessary consequence of 
this physical phenomenon, and of the limited and very small 
number of possible arrangements, that we get similar appearances, — 
capable of representation by the same diagram, in the most 
diverse fields of biology*. 


Fig. 159. 


Among the published figures of embryonic stages and other 
cell aggregates, we only discern these little intermediate partitions 
in cases where the investigator has drawn carefully just what lay 
before him, without any preconceived notions as to radial or other 
symmetry; but even in other cases we can generally recognise, 
without much difficulty, what the actual arrangement was whereby 
the cell-walls met together in equilibrium. I have a strong sus- 
picion that a leaning towards Sachs’s Rule, that one cell-wall tends 
to set itself at right angles to another cell-wall (a rule whose strict 
limitations, and narrow range of application, we have already 


* Even in a Protozoon (Huglena viridis), when kept alive under artificial com- 
pression, Ryder found a process of cell-division to occur which he compares to 
the segmenting blastoderm of a fish’s egg, and which corresponds in its essential 
features with that here described. Contrib. Zool. Lab. Univ. Pennsylvania, 1, 
pp. 37-50, 1893. 


vu] THE SEGMENTATION: OF THE EGG 377 


considered) is responsible for many inaccurate or incomplete 
representations of the mutual arrangement of aggregated cells. 
In the accompanying series of figures (Figs. 160-167) I have 


eS G19 OF 


Fig. 160. Segmenting egg Fig. 161. re views of segmenting egg of Cynthia 
of Trochus. (After Robert.) partita. (After Conklin.) 


a b 


Fig. 162. (a) Section of apical cone of Salvinia. (After Pringsheim~*.) 
(b) Diagram of probable actual arrangement. 


i : 
Fig. 163. Egg of Pyrosoma. Fig. 164. Egg of Hchinus, segmenting 
(After Korotneff). under pressure. (After Driesch.) 


* This, like many similar figures, is manifestly drawn under the influence of 
Sachs’s theoretical views, or assumptions, regarding orthogonal trajectories, coaxial 
circles, confocal ellipses, etc. 


378 THE FORMS OF TISSUES \ [CH. 


set forth a few aggregates of eight cells, mostly from drawings of 
segmenting eggs. In some cases they shew clearly the manner 
in which the cells meet one another, always at angles of 120°, 


oO 
0/O/9/0 O ofo 
a b 


Fig. 165. (a) Part of segmenting egg of Cephalopod (after Watase); 
(b) probable actual arrangement. 


Fig. 166. (a) Egg of Echinus; (b) do. of Nereis, under pressure. (After 
Driesch). 


a b 
Fig. 167. (a) Egg of frog, under pressure (after Roux); (6) probable 
actual arrangement. 
and always with the help of five intermediate boundary walls 
within the eight-celled system; in other cases I have added a 
slightly altered drawing, so as to shew, with as little change as 


vit] - THE SEGMENTATION OF THE EGG 379 


possible, the arrangement of boundaries which probably actually 
existed, and gave rise to the appearance which the observer drew. 
These drawings may be compared with the various diagrams of 
Fig. 158, in which some seven out of the possible thirteen arrange- 
ments of five intermediate partitions (for a system of eight cells) 
have been already set forth. 

It will be seen that M. Robert-Tornow’s figure of the segmenting 
ege of Trochus (Fig. 160) clearly shews the cells grouped after the 
fashion of Fig. 158, a. In hke manner, Mr Conklin’s figure of the 
ascidian egg (Cynthia) shews equally clearly the arrangement g. 

A sea-urchin egg, segmenting under pressure, as figured by 
Driesch, scarcely requires any modification of the drawing to 
appear as a diagram of the type d. Turning for a moment to a 
botanical illustration, we have a figure of Pringsheim’s shewing an 
eight-celled stage in the apex of the young cone of Salvinia; it 
is in all probability referable, as in my modified diagram, to type 
c. Beside it is figured a very different object, a segmenting egg 
of the Ascidian Pyrosoma, after Korotneff; it may be that this 
also is to be referred to type c, but I think it is more easily referable 
to type b. For there is a difference between this diagram and 
that of Salvinia, in that here apparently, of the pairs of lateral 
cells, the upper and the lower cell are alternately the larger, while 
in the diagram of Salvinia the lower lateral cells both appear much 
larger than the upper ones; and this difference tallies with the 
appearance produced if we fill in the eight cells according to the 
type b or the type c. In the segmenting cuttlefish egg, there 
is again a slight dubiety as to which type it should be referred to, 
but it is in all probability referable, like Driesch’s Echinus egg, 
to d. Lastly, I have copied from Roux a curious figure of the 
egg of Rana esculenta, viewed from the animal pole, which appears 
to me referable, in all probability, to type g. Of type f, in which 
the five partitions form a figure with four re-entrant angles, that 
is to say a figure representing the five sides of a hexagon, I have 
found no examples among segmenting eggs, and that arrange- 
ment in all probability is a very unstable one. 


It is obvious enough, without more ado, that these phenomena 
are in the strictest and completest way common to both plants 


380 THE FORMS OF TISSUES cH. 


and animals. In other words they tally with, and they further 
extend, the general and fundamental conclusions laid down by 
Schwann, in his Mikroskopische Untersuchungen iiber die Ueberein- 
stimmung in der Struktur und dem Wachsthum der Thiere und 
Pflanzen. 

But now that we have seen how a certain limited number of 
types of eight-celled segmentation (or of arrangements of eight 
cell-partitions) appear and reappear, here and there, throughout 
the whole world of organisms, there still remains the very important 
question, whether in each particular organism the conditions are 
such as to lead to one particular arrangement being predominant, 
characteristic, or even invariable. In short, isa particular arrange- 
ment of cell-partitions to be looked upon (as the published figures 
of the embryologist are apt to suggest) as a specific character, or 
at least a constant or normal character, of the particular organism ? 
The answer to this question is a direct negative, but it is only in 
the work of the most careful and accurate observers that we find 
it revealed. Rauber (whom we have more than once had occasion 
to quote) was one of those embryologists who recorded just what 
he saw, without prejudice or preconception; as Boerhaave said 
of Swammerdam, quod vidit id asserwit. Now Rauber has put on 
record a considerable number of variations in the arrangement of 
the first eight cells, which form a discoid surface about the dorsal 
(or “animal”’) pole of the frog’s egg. In a certain number of 
cases these figures are identical with one another in type, identical 
(that is to say) save for slight differences in magnitude, relative 
proportions, or orientation. But I have selected (Fig. 168) six 
diagrammatic figures, which are all essentially different, and these 
diagrams seem to me to bear intrinsic evidence of their accuracy : 
the curvatures of the partition-walls, and the angles at which 
they meet agree closely with the requirements of theory, and when 
they depart from theoretical symmetry they do so only to the 
shght extent which we should naturally expect in a material and 
imperfectly homogeneous system”. 

* Such preconceptions as Rauber entertained were all in a direction likely to 
lead him away from such phenomena as he has faithfully depicted. Rauber had 
no idea whatsoever of the principles by which we are guided in this discussion, 


nor does he introduce at all the analogy of surface-tension, or any other purely 
physical concept. But he was deeply under the influence of Sachs’s rule of rect- 


vit] THE SEGMENTATION OF THE EGG 381 


Of these six illustrations, two are exceptional. In Fig. 168, 5, 
we observe that one of the eight cells is surrounded on all sides 
by the other seven. This is a perfectly natural condition, and 
represents, like the rest, a phase of partial or conditional equili- 
brium. But it is not included in the series we are now considering, 
which is restricted to the case of eight cells extending outwards 
to a common boundary. The condition shewn in Fig. 168, 6, is 
again peculiar, and is probably rare; but it is included under the 
cases considered on p. 312, in which the cells are not in complete 


ei) ED) 
ey Ue Ge 


Fig. 168. Various modes of grouping of eight cells, at the dorsal or 
epiblastic pole of the frog’s egg. (After Rauber.) 
fluid contact, but are separated by little droplets of extraneous 
matter; it needs no further comment. But the other four cases 
are beautiful diagrams of space-partitioning, similar to those we 
have just been considering, but so exquisitely clear that they need 
no modification, no “touching-up,” to exhibit their mathematical 
regularity. It will easily be recognised that in Fig. 168, 1 and 2, 
we have the arrangements corresponding to a and d of our diagram 
(Fig. 158): but the other two (i.e. 3 and 4) represent other of the 
thirteen possible arrangements, which are not included in that 


angular intersection; and he was accordingly disposed to look upon the configura- 
tion represented above in Fig. 168, 6, as the most typical or most primitive. 


382 THE FORMS OF TISSUES (cx. 


diagram. It would be a curious and interesting investigation to 
ascertain, in a large number of frogs’ eggs, all at this stage of 
development, the percentage of cases in which these yarious 
arrangements occur, with a view of correlating their frequency 
with the theoretical conditions (so far as they are known, or can 
be ascertained) of relative stability. One thing stands out as 
very certain indeed: that the elementary diagram of the frog’s 
ege commonly given in text-books of embryology,—in which the 
cells are depicted as uniformly symmetrical quadrangular bodies,— 
is entirely inaccurate and grossly misleading*. 

We now begin to realise the remarkable fact, which may even 
appear a startling one to the biologist, that all possible groupings’ 
or arrangements whatsoever of eight cells (where all take part in 
the surface of the group, none being submerged or wholly enveloped 
by the rest) are referable to some one or other of thirteen types or 
forms. And that all the thousands and thousands of drawings 
which diligent observers have made of such eight-celled structures, 
animal or vegetable, anatomical, histological or embryological, are 
one and all representations of some one or another of these thirteen 
types :—or rather indeed of somewhat less than the whole thirteen, 
for there is reason to believe that, out of the total number of 
possible groupings, a certain small number are essentially unstable, 
and have at best, in the concrete, but a transitory and evanescent 
existence. 


Before we leave this subject, on which a vast deal more might 
be said, there are one or two points which we must not omit to 
consider. Let us note, in the first place, that the appearance 
which our plane diagrams suggest of inequality of the several 
cells is apt to be deceptive; for the differences of magnitude 
apparent in one plane may well be, and probably generally are, 
balanced by equal and opposite differences in another. Secondly, 
let us remark that the rule which we are considering refers only 


* Cf. Rauber, Neue Grundlage z. K. der Zelle, Morph. Jahrb. vim, 1883, pp. 273, 
274: 

“Ich betone noch, dass unter meinen Figuren diejenige gar nicht enthalten ist, 
welche zum Typus der Batrachierfurchung gehérig am meisten bekannt ist....Es 
haben so ausgezeichnete Beobachter sie als vorhanden beschrieben, dass es mir 
nicht einfallen kann, sie tiberhaupt nicht anzuerkennen.”’ 


vur] THE PARTITIONING OF SPACE 383 


to angles, and to the number, not to the length of the intermediate 
partitions ; itis toa great extent by variations in the length of these 
that the magnitudes of the cells may be equalised, or otherwise 
balanced, and the whole system brought into equilibrium. Lastly, 
there is a curious point to consider, in regard to the number of 
actual contacts, in the various cases, between cell and cell. If we 
inspect the diagrams in Fig. 169 (which represent three out of our 
thirteen possible arrangements of eight cells) we shall see that, in 
the case of type b, two cells are each in contact with two others, 
two cells with three others, and four cells each with four other cells. 
In type a four cells are each in contact with two, two with four, 
and two with five. In type f, two are in contact with two, four 
with three, and one with no less than seven. In all cases the 


HO Gly AO 
OY EL IES 


b ne 
Fig. 169. 


number of contacts is twenty-six in all; or, in other words, there 
are thirteen internal partitions, besides the eight peripheral walls. 
For it is easy to see that, in all cases of n cells with a common 
external boundary, the number of internal partitions is 2n — 3; 
or the number of what we call the internal or interfacial contacts 
is 2(2n — 3). But it would appear that the most stable arrange- 
ments are those in which the total number of contacts is most 
evenly divided, and the least stable are those in which some one 
cell has, as in type f, a predominant number of contacts. In a 
well-known series of experiments, Roux has shewn how, by means 
of oil-drops, various arrangements, or aggregations, of cells can 
be simulated ; and in Fig. 170 I shew a number of Roux’s figures, 
and have ascribed them to what seem to be their appropriate 
“types”? among those which we have just been considering; but 


384 THE FORMS OF TISSUES [CH. 


it will be observed that in these figures of Roux’s the drops are not 
always in complete contact, a little air-bubble often keeping them 
apart at their apical junctions, so that we see the configuration 
towards which the system is tending rather than that which it has 
fully attained*. The type which we have called f was found by 
Roux to be unstable, the large (or apparently large) drop a’’ 
quickly passing into the centre of the system, and here taking up 
a position of equilibrium in which, as usual, three cells meet 
throughout in a point, at equal angles, and in which, in this case, 
all the cells have an equal number of “interfacial” contacts. 


a 


1 2 8 
5 
oy 

4 f 6 


Fig. 170. Aggregations of oil-drops. (After Roux.) Figs. 4-6 represent 
successive changes in a single system. 


We need by no means be surprised to find that, in such arrange- 
ments, the commonest and most stable distributions are those in 
which the cell-contacts are distributed as uniformly as possible 
between the several cells. We always expect to find some such 
tendency to equality in cases where we have to do with small 
oscillations on either side of a symmetrical condition. 

* Roux’s experiments were performed with drops of paraffin suspended in 


dilute alcohol, to which a little calcium acetate was added to form a soapy pellicle 
over the drops and prevent them from reuniting with one another. 


vut| THE PARTITIONING OF SPACE 385 


The rules and principles which we have arrived at from the 
point of view of surface tension have a much wider bearing than is 
at once suggested by the problems to which we have applied them ; 
for in this elementary study of the cell-boundaries in a segmenting 
egg or tissue we are on the verge of a difficult and important 
subject in pure mathematics. It is a subject adumbrated by 
Leibniz, studied somewhat more deeply by Euler, and greatly 
developed of recent years. It is the Geometria Situs of Gauss, the 
Analysis Situs of Riemann, the Theory of Partitions of Cayley, 
and of Spatial Complexes of Listing*. The crucial point for the 
biologist to comprehend is, that in a closed surface divided into 
a number of faces, the arrangement of all the faces, lines and 
points in the system is capable of analysis, and that, when the 
number of faces or areas is small, the number of possible arrange- 
ments is small also. This is the simple reason why we meet in 
such a case as we have been discussing (viz. the arrangement of 
a group or system of eight cells) with the same few types recurring 
again and again in all sorts of organisms, plants as well as animals, 
and with no relation to the lines of biological classification: and. 
why, further, we find similar configurations occurring to mark 
the symmetry, not of cells merely, but of the parts and organs of 
entire animals. The phenomena are not “functions,” or specific 
characters, of this or that tissue or organism, but involve general 
principles which lie within the province of the mathematician. 


The theory of space-partitioning, to which the segmentation 
of the egg gives us an easy practical introduction, is illustrated in 
much more complex ways in other fields of natural history. A 
very beautiful but immensely complicated case is furnished by 
the “venation”’ of the wings of insects. Here we have sometimes 
(as in the dragon-flies), a general reticulum of small, more or less 
hexagonal “cells”: but in most other cases, in flies, bees, butter- 
flies, etc., we have a moderate number of cells, whose partitions 
always impinge upon one another three by three, and whose 
arrangement, therefore, includes of necessity a number of small 
intermediate partitions, analogous to our polar furrows. I think 

* Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., Trans. R. S. EH. xxv1, 
p. 9, 1870. 


T. G. 25 


386 THE FORMS OF TISSUES » feats 


that a mathematical study of these, including an investigation of 
the “deformation” of the wing (that is to say, of the changes in 
shape and changes in the form of its “cells” which it undergoes 
during the life of the individual, and from one species to another) 
would be of great interest. In very many cases, the entomologist 
relies upon this venation, and upon the occurrence of this or that ° 
intermediate vein, for his classification, and therefore for his 
hypothetical phylogeny of particular groups; which latter pro- 
cedure hardly commends itself to the physicist or the mathe- 
matician. 

Another case, geometrically akin but biologically very 
different, is to be found in the little diatoms of the genus Astero- 
lampra, and their immediate congeners*. In Asterolampra we 


A B Cc 


Fig. 171. (A) Asterolampra marylandica, Ehr.; (B, C) A. variabilis, Grev. 
(After Greville. ) 


have a little disc, in which we see (as it were) radiating spokes of 
one material, alternating with intervals occupied on the flattened 
wheel-like disc by another (Fig. 171). The spokes vary in number, 
but the general appearance is in a high degree suggestive of the 
Chladni figures produced by the vibration of a circular plate. 
The spokes broaden out towards the centre, and interlock by 
visible junctions, which obey the rule of triple intersection, and 
accordingly exemplify the partition-figures with which we are 
dealing. But whereas we have found the particular arrangement 
in which one cell is in contact with all the rest to be unstable, 
according to Roux’s oil-drop experiments, and to be conspicuous 


* See Greville, K. R., Monograph of the Genus Asterolampra, Q.J.M.S. vii, 
(Trans.), pp. 102-124, 1860; cf. cbid. (n.s.), 1, pp. 41-55, 1862. 


viii] THE PARTITIONING OF SPACE 387 


by its absence from our diagrams of segmenting eggs, here in 
Asterolampra, on the other hand, it occurs frequently, and is 
indeed the commonest arrangement* (Fig. 171, B). In all proba- 
bility, we are entitled to consider this marked difference natural 
enough. For we may suppose that in Asterolampra (unlike the 
case of the segmenting egg) the tendency is to perfect radial 
symmetry, all the spokes emanating from a point in the centre: 
such a condition would be eminently unstable, and would break 
down under the least asymmetry. A very simple, perhaps the 
simplest case, would be that one single spoke should differ slightly 
from the rest, and should so tend to be drawn in amid the others, 
these latter remaining similar and symmetrical among themselves. 
Such a configuration would be vastly less unstable than the 
original one in which all the boundaries meet in a point; and the 
fact that further progress is not made towards other configurations 
of still greater stability may be sufficiently accounted for by 
viscosity, rapid solidification, or other conditions of restraint. 
A perfectly stable condition would of course be obtained if, as in 
the case of Roux’s oil-drop (Fig. 170, 6), one of the cellular spaces 
passed into the centre of the system, the other partitions radiating 
outwards from its circular wall to the periphery of the whole 
system. Precisely such a condition occurs among our diatoms; 
but when it does so, it is looked 

upon as the mark and characterisa- 

tion of the allied genus Arachnoid- 

iSCus. 


In a diagrammatic section of 
an Alcyonarian polype (Fig. 172), 
we have eight chambers set, sym- 
metrically, about a ninth, which 
constitutes the “stomach.” In this 
arrangement there is no difficulty, 
_for it is obvious that, throughout 
the system, three boundaries meet 
(in plane section) in a poimt. In many corals we have as 


Fig. 172. Section of Alcyonarian 
polype. 


* The same is true of the insect’s wing; but in this case I do not hazard a 
conjectural explanation. 


25—2 


388 THE FORMS OF TISSUES [CH. 


simple, or even simpler conditions, for the radiating calcified 
partitions either converge upon a central chamber, or fail to 
meet it and end freely. But in a few cases, the partitions or 
“septa” converge to meet one another, there being no central 
chamber on which they may impinge; and here the manner in 
which contact is effected becomes complicated, and involves 
problems identical with those which we are now studying. 

In the great majority of corals we have as simple or even 
simpler conditions than those of Aleyonium; for as a rule the 
calcified partitions or septa of the coral 
either converge upon a central chamber 
(or central “columella’’), or else fail to 
meet it and end freely. In the latter 
case the problem of space-partitioning 
does not arise; in the former, however 
numerous the septa be, their separate 
contacts with the wall of the central 


Fig. 173. Heterophyllia angu- chamber comply with our fundamental 


lata. (After Nicholson.) rule according to which three lines and 


no more meet in a point, and from this simple and symmetrical 
arrangement there is little tendency to variation. But in a few 
cases, the septal partitions converge to, meet one another, there 
being no central chamber on which they may impinge; and here 
the manner in which contact is effected becomes complicated, and 
involves problems of space-partitioning identical with those which 
we are now studying. In the genus Heterophyllia and in a few 
allied forms we have such conditions, and students of the Coelen- 
terata have found them very puzzling. McCoy*, their first 
discoverer, pronounced these corals to be “totally unlike” any 
other group, recent or fossil; and Professor Martin Duncan, 
writing a memoir on Heterophyllia and its allies}, described them 
as “paradoxical in their anatomy.” . 

The simplest or youngest Heterophylliae known have six septa 
(as in Fig. 174, a); im the case figured, four of these septa are 
conjoined two and two, thus forming the usual triple junctions 
together with their intermediate partition-walls: and in the 


* Ann. Mag. N. H. (2), mH, p. 126, 1849. 
t Phil. Trans. civi, pp. 643-656, 1867. 


vur] THE PARTITIONING OF SPACE 389 


case of the other two we may fairly assume that their proper 
and original arrangement was that of our type 6b (Fig. 158), 
though the central intermediate partition has been crowded out 
by partial coalescence. When with increasing age the septa 
become more numerous, their arrangement becomes exceedingly 
variable; for the simple reason that, from the mathematical 
point of view, the number of possible arrangements, of 10, 12 
or more cellular partitions in triple contact, tends to increase 
with great rapidity, and there is little to choose between many 


Fig. 174. Heterophyllia sp. (After Martin Duncan.) 


of them in regard to symmetry and equilibrium. But while, 
mathematically speaking, each particular case among the multi- 
tude of possible cases is an orderly and definite arrangement, 
from the purely biological point of view on the other hand no 
law or order is recognisable; and so McCoy described the genus 
as being characterised by the possession of septa “destitute of any 
order of arrangement, but irregularly branching and coalescing in 
their passage from the solid external walls towards some indefinite 
point near the centre where the few main lamellae irregularly 
anastomose.” 


390 THE FORMS OF TISSUES [CH. 


In the two examples figured (Fig. 174), both comparatively 
simple ones, it will be seen that, of the main chambers, one is in 
each case an unsymmetrical one; that is to say, there is one 
chamber which is in contact with a greater number of its neighbours 
than any other, and which at an earlier stage must have had 
contact with them all; this was the case of our type /, in the 
eight-celled system (Fig. 158). Such an asymmetrical chamber 
(which may occur in a system of any number of cells greater than 
six), constitutes what is known to students of the Coelenterata as 
a “fossula’’; and we may recognise it not only here, but also in 
Zaphrentis and its allies, and in a good many other corals besides. 
Moreover certain corals are described as having more than one 
fossula: this appearance being naturally produced under certain 
of the other asymmetrical variations of normal space-partitioning. 
Where a single fossula occurs, we are usually told that it is a 
symptom of “bilaterality”; and this is in turn interpreted as 
an indication of a higher grade of organisation than is implied 
in the purely “radial symmetry” of the commoner types of coral. 
The mathematical aspect of the case gives no warrant for this 
interpretation. 

Let us carefully notice (lest we run the risk of confusing two 
distinct problems) that the space-partitioning of Heterophylha 
by no means agrees with the details of that which we have studied 
in (for instance) the case of the developing disc of Erythrotrichia : 
the difference simply being that Heterophylha illustrates the 
general case of cell-partitioning as Plateau and Van Rees studied 
it, while in Erythrotrichia, and in our other embryological and 
histological instances, we have found ourselves justified in making 
the additional assumption that each new partition divided a cell 
into co-equal parts. No such law holds in Heterophylla, whose 
case is essentially different from the others: inasmuch as the 
chambers whose partition we are discussing in the coral are mere 
empty spaces (empty save for the mere access of sea-water); while 
in our histological and embryological instances, we were speaking 
of the division of a cellular unit of living protoplasm. Accordingly, 
among other differences, the “transverse” or “periclnal” parti- 
tions, which were bound to appear at regular intervals and in 
definite positions, when co-equal bisection was a feature of the 


Vit | THE PARTITIONING OF SPACE 391 


case, are comparatively few and irregular in the earlier stages of 
Heterophyllia, though they begin to appear in numbers after the 
main, more or less radial, partitions have become numerous, and 
when accordingly these radiating partitions come to bound narrow 
and almost parallel-sided interspaces; then it is that the transverse 
or periclinal partitions begin to come in, and form what the student 
of the Coelenterata calls the “dissepiments” of the coral. We 
need go no further into the configuration and anatomy of the 
corals; but it seems to me beyond a doubt that the whole question 
of the complicated arrangement of septa and dissepiments through- 
out the group (including the curious vesicular or bubble-like 
tissue of the Cyathophyllidae and the general structural plan of 


Fig. 175. Diagrammatic section of a Ctenophore (Hucharis). 


the Tetracoralla, such as Streptoplasma and its allies) is well 
worth investigation from the physical and mathematical point of 
view, after the fashion which is here slightly adumbrated. 


The method of dividing a circular, or spherical, system into 
eight parts, equal as to their areas but unequal in their peripheral 
boundaries, is probably of wide biological application; that is to 
say, without necessarily supposing it to be rigorously followed, the 
typical configuration which it yields seems to recur again and 
again, with more or less approximation to precision, and under 
widely different circumstances. I am inclined to think, for instance, 
that the unequal division of the surface of a Ctenophore by its 


392 THE FORMS OF TISSUES [CH. 


meridian-like ciliated bands is a case in point (Fig. 175). Here, if we 
imagine each quadrant to be twice bisected by a curved anticline, 
we shall get what is apparently a close approximation to the actual 
position of the ciliated bands. The case however is complicated 
by the fact that the sectional plan of the organism is never quite 
circular, but always more or less elliptical. One point, at least, 
is clearly seen in the symmetry of the Ctenophores; and that is 
that the radiating canals which pass outwards to correspond in 
position with the ciliated bands, have no common centre, but 
diverge from one another by repeated bifurcations, in a manner 
comparable to the conjunctions of our cell-walls. 

In like manner I am inclined to suggest that the same principle 
may help us to understand the apparently complex arrangement 


B 


Fig. 176. Diagrammatic arrangement of partitions, represented by skeletal 
rods, in larval Echinoderm (Ophiura). 


of the skeletal rods of a larval Echinoderm, and the very complex 
conformation of the larva which is brought about by the presence 
of these long, slender skeletal radii. 

In Fig. 176 I have divided a circle into its four quadrants, and 
have bisected each quadrant by a circular arc (BC), passing from 
radius to periphery, as in the foregoing cases of cell-division; and 
I have again bisected, in a similar way, the triangular halves of 
each quadrant (DD). I have also inserted a small circle in the 
middle of the figure, concentric with the large one. If now we 
imagine those lines in the figure which I have drawn black to be 
replaced by solid rods we shall have at once the frame-work of an 
Ophiurid (Pluteus) larva. Let us imagine all these arms to be 


vir] THE PARTITIONING OF SPACE 393 


bent symmetrically downwards, so that the plane of the paper is 
transformed into a spheroidal surface, such as that of a hemisphere, 
or that of a tall conical figure with curved sides; let a membrane 
be spread, umbrella-like, between the outstretched skeletal rods, 
and let its margin loop from rod to rod in curves which are possibly 
catenaries, but are more probably portions of an “elastic curve,’ 
and the outward resemblance to a Pluteus larva is now complete. 
By various slight modifications, by altering the relative lengths 
of the rods, by modifying their curvature or by replacing the curved 
rod by a tangent to itself, we can ring the changes which lead us 
from one known type of Pluteus to another. The case of the 
Bipinnaria larvae of Echinids is certainly analogous, but it be- 
comes very much more complicated; we have to do with a more 


Fig. 177. Pluteus-larva of Ophiurid. 


complex partitioning of space, and I confess that | am not yet 
able to represent the more complicated forms in so simple a way. 


There are a few notable exceptions (besides the various un- 
equally segmenting eggs) to the general rule that in cell-division 
the mother-cell tends to divide into equal halves; and one of these 
exceptional cases is to be found in connection with the develop- 
ment of “stomata” in the leaves of plants. The epidermal cells 
by which the leaf is covered may be of various shapes; sometimes, 
as in a hyacinth, they are oblong, but more often they have an 
irregular shape in which we can recognise, more or less clearly, 
a distorted or imperfect hexagon. In the case of the oblong cells, 
a transverse partition will be the least possible, whether the cell 
be equally or unequally divided, unless (as we have already seen 


! 


394 THE FORMS OF TISSUES fou. 


the space to be cut off be a very small one, not more than about 
three-tenths the area of a square based on the short side of the 
original rectangular cell. As the portion usually cut off is not 
nearly so small as this, we get the form of partition shewn in 


Fig. 178. Diagrammatic development of Stomata in Sedum. (Cf. fig. in 
Sachs’s Botany, 1882, p. 103.) 


Fig. 179, and the cell so cut off is next bisected by a partition at 
right angles to the first; this latter partition splits, and the two 
last-formed cells constitute the so-called “guard-cells” of the 
stoma. In other cases, as in Fig. 178, there will come a point 
where the minimal partition necessary to cut off the required 
fraction of the cell-content is no longer a transverse one, but is 
a portion of a cylindrical wall (2) cutting off one corner of the 

mother-cell. The cell so cut off 


is now a certain segment of a 
2b circle, with an arc of approxi- 
mately 120°; and its next division 
will be by means of a curved wall 


cutting it into a triangular and 

a quadrangular portion (3). The 

triangular portion will continue to 

divide in a similar way (4, 5), 

Fig. 179. Diagrammatic development and at length (for a reason which 
of stomata in Hyacinth. is not yet clear) the partition wall 


VIIt} THE DEVELOPMENT OF STOMATA 395 


between the new-formed cells splits, and again we have the 
phenomenon of a “stoma” with its attendant guard-cells. In 
Fig. 179 are shewn the successive stages of division, and the 
changing curvatures of the various walls which ensue as each 
subsequent partition appears, introducing a new tension into the 
system. 

It is obvious that in the case of the oblong cells of the epidermis 
in the hyacinth the stomata will be found arranged in regular rows, 
while they will be irregularly distributed over the surface of the 
leaf in such a case as we have depicted in Sedum. 

While, as I have said, the mechanical cause of the split which 
constitutes the orifice of the stoma is not quite clear, yet there 
can be little or no doubt that it, like the rest of the phenomenon, 
is related to surface tension. It might well be that it is directly 
due to the presence underneath this portion of epidermis of the 
hollow air-space which the stoma is apparently developed “for 
the purpose” of communicating with; this air-surface on both. 
sides of the delicate epidermis might well cause such an alteration 
of tensions that the two halves of the dividing cell would tend to 
part company. In short, if the surface-energy in a cell-air contact 
were half or less than half that in a contact between cell and cell, 
then it is obvious that our partition would tend to split, and give 
us a two-fold surface in contact with air, instead of the original 
boundary or interface between one cell and the other. In Professor 
‘Macallum’s experiments, which we have briefly discussed in our 
short chapter on Adsorption, it was found that large quantities 
of potassium gathered together along the outer walls of the guard- 
cells of the stoma, thereby indicating a low surface-tension along 
these outer walls. The tendency of the guard-cells to bulge 
outwards is so far explained, and it is possible that, under the 
existing conditions of restraint, we may have here a force tending, 
or helping, to split the two cells asunder. It is clear enough, 
however, that the last stage in the development of a stoma, is, 
from the physical point of view, not yet properly understood. 


In all our foregoing examples of the development of a “tissue” 
we have seen that the process consists in the successive division 
of cells, each act of division being accompanied by the formation 


396 THE FORMS OF TISSUES [CH. 


of a boundary-surface, which, whether it become at once a solid 
or semi-solid partition or whether it remain semi-fluid, exercises 
in all cases an effect on the position and the form of the boundary 
which comes into being with the next act of division. In contrast 
to this general process stands the phenomenon known as “free 
cell-formation,”’ in which, out of a common mass of protoplasm, 
a number of separate cells are simultaneously, or all but simu!- 
taneously, differentiated. In a number of cases it happens that, 
to begin with, a number of “mother-cells” are formed simul- 
taneously, and each of these divides, by two successive divisions, 


Fig. 180. Various pollen-grains and spores (after Berthold, Campbell, Goebel 
and others). (1) Epilobium; (2) Passiflora; (3) Neottia; (4) Periploca 
graeca; (5) Apocynum; (6) Erica; (7) Spore of Osmunda; (8) Tetraspore of 
Callithamnion. 


into four “daughter-cells.”” These daughter-cells will tend to group 
themselves, just as would four soap-bubbles, into a “tetrad,” the 
four cells corresponding to the angles of a regular tetrahedron. 
For the system of four bodies is evidently here in perfect symmetry ; 
the partition-walls and their respective edges meet at equal 
angles: three walls everywhere meeting in an edge, and the four 
edges converging to a point in the geometrical centre of the 
system. This is the typical mode of development of pollen- 
grains, common among Monocotyledons and all but universal . 
among Dicotyledonous plants. By a loosening of the surrounding 
tissue and an expansion of the cavity, or anther-cell, in which 


vit] THE SHAPES OF POLLEN-GRAINS 397 


they lie, the pollen-grains afterwards fall apart, and their in- 
dividual form will depend upon whether or no their walls have 
solidified before this hberation takes place. 
For if not, then the separate grains will be 
free to assume a spherical form as a con- 
sequence of their own individual and un- © 
restricted growth; but if they become solid 
or rigid prior to the separation of the Go 
tetrad, then they will conserve more or less 
completely the plane interfaces and sharp  Fig.181. Dividingspore 
angles of the elements of the tetrahedron. Cee pany CENCE: 
The latter is the case, for instance, in 
the pollen-grains of Epilobium (Fig. 180, 1) and in many 
others. In the Passion-flower (2) we have an intermediate 
condition: where we can still see an indication of the facets 
where the grains abutted on one another in the tetrad, but 
the plane faces have been swollen by growth into spheroidal or 
spherical surfaces. It is obvious that there may easily be cases 
where the tetrads of daughter-cells are prevented from assuming 
the tetrahedral form: cases, that is to say, where the four cells 
are forced and crushed into one plane. The figures given by 
Goebel of the development of the pollen of Neottia (3, a-e: all 
the figures referring to grains taken from a single anther), illustrate 
this to perfection; and it will be seen that, when the four cells 
he in a plane, they conform exactly to our typical diagram of the 
first four cells in a segmenting ovum. Occasionally, though the 
four cells lie in a plane, the diagram seems to fail us, for the cells 
appear to meet in a simple cross (as in 5); but here we soon 
perceive that the cells are not in complete interfacial contact, 
but are kept apart by a little intervening drop of fluid or bubble 
of air. The spores of ferns (7) develop in very much the same 
way as pollen-grains; and they also very often retain traces of 
the shape which they assumed as members of a tetrahedral figure. 
Among the “tetraspores” (8) of the Florideae, or Red Seaweeds, 
we have a phenomenon which is in every respect analogous. 
Here again it is obvious that, apart from differences in actual 
magnitude, and apart from superficial or “accidental” differences 
(referable to other physical phenomena) in the way of colour, 


398 ‘ THE FORMS OF TISSUES [CH. 


texture and minute sculpture or pattern, it comes to pass, through 
the laws of surface-tension and the principles of the geometry of 
position, that a very small number of diagrammatic figures will 
sufficiently represent the outward forms of all the tetraspores, 
four-celled pollen-grains, and other four-celled aggregates which 
are known or are even capable of existence. 


We have been dealing hitherto (save for some shght exceptions) 
with the partitioning of cells on the assumption that the system 
either remains unaltered in size or else that growth has proceeded 
uniformly in all directions. But we extend the scope of our 
enquiry very greatly when we begin to deal with unequal growth, 
with growth, that is to say, which produces a greater extension 
along some one axis than another. And here we come close in 
touch with that great and still (as I think) insufficiently appreciated 
generalisation of Sachs, that the manner in which the cells divide 
is the result, and not the cause, of the form of the dividing 
structure: that the form of the mass is caused by its growth 
as a whole, and is not a resultant of the growth of the 
cells individually considered*. Such asymmetry of growth 
may be easily imagined, and may conceivably arise from a 
variety of causes. In any individual cell, for instance, it may 
arise from molecular asymmetry of the structure of the cell-wall, 
giving it greater rigidity in one direction than another, while all 
the while the hydrostatic pressure within the cell remains constant 
and uniform. In an aggregate of cells, it may very well arise 
from a greater chemical, or osmotic, activity in one than another, 
leading to a localised increase in the fluid pressure, and to a 
corresponding bulge over a certain area of the external surface. 
It might conceivably occur as a direct result of the preceding 
cell-divisions, when these are such as to produce many peripheral 
or concentric walls in one part and few or none in another, with 
the obvious result of strengthening the common boundary wall 
and resisting the outward pressure of growth in parts where the 
former is the case; that is to say, in our dividing quadrant, if 


* Sachs, Pflanzenphysiologie ( Vorlesung xxtv), 1882; cf. Rauber, Neue Grundlage 
zur Kenntniss der Zelle, Morphol. Jahrb. vii, p. 303 seqg., 1883; E. B. Wilson, 
Cell-lineage of Nereis, Journ. of Morphology, v1, p. 448, 1892, etc. 


Vir} OR CELL-AGGREGATES 399 


its quadrangular portion subdivide by periclines, and the triangular 
portion by oblique anticlines (as we have seen to be the natural 
tendency), then we might expect that external growth would be 
more manifest over the latter than over the former areas. As 
a direct and immediate consequence of this we might expect a 
tendency for special outgrowths, or “buds,” to arise from the 
triangular rather than from the quadrangular cells; and this 
turns out to be not merely a tendency towards which theoretical 
considerations point, but a widespread and important factor in the 
morphology of the cryptogams. But meanwhile, without en- 
quiring further into this complicated question, let us simply take 
it that, if we start from such a simple case as a round cell which 
has divided into two halves, or four quarters (as the case may be), 
we shall at once get bilateral symmetry about a main axis, and 
other secondary results arising therefrom, as soon as one of the 
halves, or one of the quarters, begins to shew a rate of growth in 
advance of the others; for the more rapidly growing cell, or the 
peripheral wall common to two or more such rapidly growing cells, 
will bulge out into an ellipsoid form, and may finally extend 
into a cylinder with rounded or ellipsoid end. 

This latter very simple case is illustrated in the development 
of a pollen-tube, where the rapidly growing cell develops into the 
elongated cylindrical tube, and the slow-growing or quiescent part 
remains behind as the so-called “vegetative” cell or cells. 

Just as we have found it easier to study the segmentation of 
a circular disc than that of a spherical cell, so let us begin in the 
same way, by enquiring into the divisions which will ensue if the 
dise tend to grow, or elongate, in some one particular direction, 
instead of in radial symmetry. The figures which we shall then 
obtain will not only apply to the disc, but will also represent, in 
all essential features, a projection or longitudinal section of a solid 
body, spherical to begin with, preserving its symmetry as a solid 
of revolution, and subject to the same general laws as we have 
studied in the disc*. 

* In the following account I follow closely on the lines laid down by Berthold; 
Protoplasmamechanik, cap. vii. Many botanical phenomena identical and similar 
to those here dealt with, are elaborately discussed by Sachs in his Physiology of 


Plants (chap. xxvii, pp. 431-459, Oxford, 1887); and in his earlier papers, Ueber 
die Anordnung der Zellen in jiingsten Pflanzentheilen, and Ueber Zellenanordnung 


400 THE FORMS OF TISSUES [CH. 


(1) Suppose, in the first place, that the axis of growth lies 
symmetrically in one of the original quadrantal cells of a segmenting 
disc; and let this growing cell elongate with comparative rapidity 
before it subdivides. When it does divide, it will necessarily do 
so by a transverse partition, concave towards the apex of the 
cell: and, as further elongation takes place, the cylindrical 
structure which will be developed thereby will tend to be again 
and again subdivided by similar concave transverse partitions. 
If at any time, through this process of concurrent elongation and 
subdivision, the apical cell become equivalent to, or less than, 
a hemisphere, it will next divide by means of a longitudinal, or 


—©QBR 
BBBE 


Fig. 182. 


vertical partition; and similar longitudinal partitions will arise in 
the other segments of the cylinder, as soon as it comes about that 
their length (in the direction of the axis) is less than their breadth. 

But when we think of this structure in the solid, we at once 
perceive that each of these flattened segments of the cylinder, 
into which our cylinder has divided, is equivalent to a flattened 
circular disc; and its further division will accordingly tend to 
proceed like any other flattened disc, namely into four quadrants, 
and afterwards by anticlines and periclines in the usual way. 
und Wachsthum (Arb. d. botan. Inst. Wurzburg, 1878, 1879). But Sachs’s treat- 
ment differs entirely from that which I adopt and advocate here: his explanations 
being based on his “law” of rectangular succession, and involving complicated 


systems of confocal conics, with their orthogonally intersecting ellipses and hyper- 
bolas. 


vut] OR CELL-AGGREGATES 401 


A section across the cylinder, then, will tend to shew us precisely 
the same arrangements as we have already so fully studied in 
connection with the typical division of a circular cell into quadrants, 
and of these quadrants into triangular and quadrangular portions, 
and so on. 

But there are other possibilities to be considered, in regard to 
the mode of division of the elongating quasi-cylindrical portion, as 
it gradually develops out of the growing and bulging quadrantal 
cell; for the manner in which this latter cell divides will simply 
depend upon the form it has assumed before each successive act 
of division takes place, that is to say upon the ratio between its 
rate of growth and the frequency of its successive divisions. For, 
as we have already seen, if the growing cell attain a markedly 
oblong or cylindrical form before division ensues, then the partition 
will arise transversely to the long axis; if it be but a little more 
than a hemisphere, it will divide by an oblique partition; and if 
it be less than a hemisphere (as it may come to be after successive 
_ transverse divisions) it will divide by a vertical partition, that is 
to say by one coinciding with its axis of growth. An immense 
number of permutations and combinations may arise in this way, 
and we must confine our illustrations to a small number of cases. 
The important thing is not so much to trace out the various 
conformations which may arise, but to grasp the fundamental 
principle: which is, that the forces which dominate the form of 
each cell regulate the manner of its subdivision, that is to say 
the form of the new cells into which it subdivides; or in other 
words, the form of the growing organism regulates the form and 
number of the cells which eventually constitute it. The complex 
cell-network is not the cause but the result of the general configura- 
tion, which latter has its essential cause in whatsoever physical 
and chemical processes have led to a varying velocity of growth 
in one direction as compared with another. 

In the annexed figure of an embryo of Sphagnum we see a 
mode of development almost precisely corresponding to the 
hypothetical case which we have just described,—the case, that 
is to say, where one of the four original quadrants of the mother- 
cell is the chief agent in future growth and development. We 
see at the base of our first figure (a), the three stationary, or 


T, G. 26 


~ 402 THE FORMS OF TISSUES [CH. 


undivided quadrants, one of which has further slowly divided 
in the stage b. The active quadrant 


; has grown quickly into a cylindrical 
structure, which inevitably divides, in 
the next place, into a series of trans- 

a 5 verse partitions; and accordingly, this 
mode of development carries with it 
the presence of a single “apical cell,” 
whose lower wall is a spherical surface 

SD 


with its convexity downwards. Hach 
cell of the subdivided cylinder now ap- 
pears as a more or less flattened disc, 
whose mode of further sub-division 
we may prognosticate according to 
our former investigation, to which 
subject we shall presently return. 
(2) In the next place, still keeping to the case where only one 
of the original quadrant-cells continues to grow and develop, let 
us suppose that this growing cell falls to be divided when by 
growth it has become just a little greater than a hemisphere; it 


® Q ®& 


BRAS 


Fig. 184. 


Fig. 183. Development of 
Sphagnum. (After Campbeili.) 


will then divide, as in Fig. 184, 2, by an oblique partition, in the 
usual way, whose precise position and inclination to the base will 
depend entirely on the configuration of the cell itself, save only, 
of course, that we may have also to take into account the possibility 
of the division being into two unequal halves. By our hypothesis, 


vir] OR CELL-AGGREGATES 403 


the growth of the whole system is mainly in a vertical direction, 
which is as much as to say that the more actively growing proto- 
plasm, or at least the strongest osmotic force, will be found 
near the apex; where indeed there is obviously more external 
surface for osmotic action. It will therefore be that one of 
the two cells which contains, or constitutes, the apex which 
will grow more rapidly than the other, and which therefore will 
be the first to divide, and indeed in any case, it will usually be 
this one of the two which will tend to divide first, inasmuch 
as the triangular and not the quadrangular half is bound to 
constitute the apex*. It is obvious that (unless the act of division 
be so long postponed that the cell has become quasi-cylindrical) 
it will divide by another oblique partition, starting from, and 
running at right angles to, the first. And so division will proceed, 
by oblique alternate partitions, each one tending to 

be, at first, perpendicular to that on which it is based 

and also to the peripheral wall; but all these points of 

contact soon tending, by reason of the equal tensions 

of the three films or surfaces which meet there, to form 

angles of 120°. There will always be, in such a case, 

a single apical cell, of a more or less distinctly 

triangular form. The annexed figure of the developing 
antheridium of a Liverwort (Riccia) is a typical example 

of such a case. In Fig. 185 which represents a a as 
“gemma” of a Moss, we see just the same thing; Aiki 
with this addition, that here the lower of the two sisal 
original cells has grown even more quickly than the 

other, constituting a long cylindrical stalk, and dividing in ac- 
cordance with its shape, by means of transverse septa. 

In all such cases as these, the cells whose development we have 
studied will in turn tend to subdivide, and the manner in which 
they will do so must depend upon their own proportions; and in 
all cases, as we have already seen, there will sooner or later be 
a tendency to the formation of periclinal walls, cutting off an 
“epidermal layer of cells,” as Fig. 186 illustrates very well. 

The method of division by means of oblique partitions is a 
common one in the case of ‘growing points’; for it evidently 
* Cf. p. 369. 

26—2 


404 THE FORMS OF TISSUES [cH. 


includes all cases in which the act of cell-division does not lag 
far behind that elongation which is determined by the specific rate 
of growth. And it is also obvious that, under a common type, 


Fig. 186. Development of antheridium of Riccia. (After Campbell.) 


there must here be included a variety of cases which will, at first 
sight, present a very different appearance one from another. 
For instance, in Fig. 187 which represents a growing shoot of 
Selaginella, and somewhat less diagrammatically in the young 


Fig. 187. Section of growing shoot Fig. 188. Embryo of Jungermannia. 
of Selaginella, diagrammatic. (After Kienitz-Gerloff.) 


embryo of Jungermannia (Fig. 188), we have the appearance of 
an almost straight vertical partition running up in the axis of the 
system, and the primary cell-walls are set almost at right angles 
to it,—almost transversely, that is to say to the outer walls and 
to the long axis of the structure. We soon recognise, however, 


vur} OR CELL-AGGREGATES 405 


that the difference is merely a difference of degree. The more 
remote the partitions are, that is to say the greater the velocity 
of growth relatively to division, the less abrupt will be the 
alternate kinks or curvatures of the portions which lie in the 
neighbourhood of the axis, and the more will these portions 
appear to constitute a single unbroken wall. 

(3) But an appearance nearly, if not quite, indistinguishable 
from this may be got in another way, namely, when the original 
growing cell is so nearly hemispherical that it is actually divided 
by a vertical partition, into two quadrants; and from this vertical 
partition, as it elongates, lateral partition-walls will arise on either 
side. And by the tensions exercised by these, the vertical partition 
will be bent into little portions set at 120° one to another, and the 


PARE E 


Fig. 189. 


whole will come to look just like that which, in the former case, 
was made up of portions of many successive oblique partitions. 


Let us now, in one or two cases, follow out a little further the 
stages of cell-division whose beginning we have studied in the last 
paragraphs. In the antheridium of Riccia, after the successive 
oblique partitions have produced the longitudinal series of cells 
shewn in Fig. 186, it is plain that the next partitions will arise 
periclinally, that is to say parallel to the outer wall, which in 
this particular case represents the short axis of the oblong cells. 
The effect is at once to produce an epidermal layer, whose cells 
will tend to subdivide further by means of partitions perpendicular 
to the free surface, that is to say crossing the flattened cells by 
their shortest diameter. The inner mass, beneath the epidermis, 
consists of cells which are still more or less oblong, or which become 


406 THE FORMS OF TISSUES [CH. 


definitely so in process of growth; and these again divide, parallel 
to their short axes, into squarish cells, which as usual, by the 
mutual tension of their walls, become hexagonal, as seen in a plane 
section. There is a clear distinction, then, in form as well as in 
position, between the outer covering-cells and those which lie 
within this envelope; the latter are reduced to a condition which 
merely fulfils the mechanical function of a protective coat, while 
the former undergo less modification, and give rise to the actively 
living, reproductive elements. 

In Fig. 190 is shewn the development of the sporangium of a 
fern (Osmunda). We may trace here the common phenomenon 
of a series of oblique partitions, built alternately on one another, 


Fig. 190. Development of sporangium of Osmunda. (After Bower.) 


and cutting off a conspicuous triangular apical cell. Over the 
whole system an epidermal layer has been formed, in the manner 
we have described; and in this case it covers the apical cell also, 
owing to the fact that it was of such dimensions that, at one stage 
of growth, a periclinal partition wall, cutting off its outer end, 
was indicated as of less area than an anticlinal one. This periclinal 
wall cuts down the apical cell to the proportions, very nearly, 
of an equilateral triangle, but the solid form of the cell is obviously 
that of a tetrahedron with curved faces; and accordingly, the 
least possible partitions by which further subdivision can be 
effected will run successively parallel to its four sides (or its three 
sides when we confine ourselves to the appearances as seen in 


vir] OR CELL-AGGREGATES 407 


section). The effect, as seen in section, is to cut off on each side 
a characteristically flattened cell, oblong as seen in section, still 
leaving a triangular (or strictly speaking, a tetrahedral) one in 
the centre. The former cells, which constitute no specific structure 
or perform no specific physiological function, but which merely 
represent certain directions in space towards which the whole 
system of partitioning has gradually led, are called by botanists 
the “tapetum.” The active growing tetrahedral cell which lies 
between them, and from which'in a sense every other cell in the 
system has been either directly or indirectly segmented off, still 
manifests, as it were, its vigour and activity, and now, by 
internal subdivision, becomes the mother-cell of the spores. 


In all these cases, for simplicity’s sake, we have merely con- 
sidered the appearances presented in a single, longitudinal, plane 
of optical section. But it is not difficult to interpret from these 
appearances what would be seen in another plane, for instance 
in a transverse section. In our first example, for instance, that 
of the developing embryo of Sphagnum (Fig. 183), we can see that, 
at appropriate levels, the cells of the original cylindrical row have 
divided into transverse rows of four, and then of eight cells. We 
may be sure that the four cells represent, approximately, quadrants 
of a cylindrical disc, the four cells, as usual, not meeting in a point, 
but intercepted by a small intermediate partition. Again, where 
we have a plate of eight cells, we may well imagine that the eight 
octants are arranged in what we have found to be the way 
naturally resulting from the division of four quadrants, that is to 
say into alternately triangular and quadrangular portions; and 
this is found by means of sections to be the case. The accompany- 
ing figure is precisely comparable to our previous diagrams of the 
arrangement of an aggregate of eight cells in a dividing disc, save 
only that, in two cases, the cells have already undergone a further 
subdivision. 

It follows in like manner, that in a host of cases we meet with 
this characteristic figure, in one or other of its possible, and 
strictly limited, variations,—in the cross sections of growing 
embryonic structures, just as we have already seen that it appears 
in a host of cases where the entire system (or a portion of its 


408 THE FORMS OF TISSUES [cH. 


surface) consists of eight cells only. For example, in Fig. 191, 


A 


Fig. 191. (A, B,) Sections of younger and older embryos of Phascum; 
(C) do. of Adiantum. (After Kienitz-Gerloff.) 


we have it again, in a section of a young embryo of a moss (Phas- 
cum), and in a section of an embryo of a fern (Adiantum). In 
Fig. 192 shewing a section through 
a growing frond of a sea-weed 
(Girardia) we have a case where 
the partitions forming the eight 
octants have conformed to the 
, usual type; but instead of the 
usual division by periclines of the 
four quadrangular spaces, these 
latter are dividing by means of 
oblique septa, apparently owing 
to the fact that the cell is not 
dividing into two equal, but into Fig. 192. Section through frond 
two unequal portions. In this last eee sphacelaria:© ai 
figure we have a peculiar look of 
stiffness or formality, such that it appears at first to bear little 
resemblance to the rest. The explanation is of the simplest. 
The mode of partitioning differs little (except to some slight 
extent in the way already mentioned) from the normal type; 
but in this case the partition walls are so thick and become 
so quickly comparatively solid and rigid, that the secondary 
curvatures due to their successive mutual tractions are here 
imperceptible. 

A curious and beautiful case, apparently aberrant but which 
would doubtless be found conforming strictly to physical laws, if 


ViIt] OR CELL-AGGREGATES 409 


only we clearly understood the actual conditions, is indicated in 
the development of the antheridium 

of a fern, as described by Strasbiirger. 

Here the antheridium develops from 

a single cell, whose form has grown / 
to be something more than a hemi- 

sphere; and the first partition, instead a 

of stretching transversely across the neh 

cell, as we should expect it to do if ey 

the cell were actually spherical, has X 
as it were sagged down to come in Fig. 193. Development of anthe- 
contact with the base,andsotodevelop — tidium of Pteris. (After 

: He - Strasbiirger. ) 

into an annular partition, running 

round the lower margin of the cell. The phenomenon is akin to that 
cutting off of the corner of a cubical cell by a spherical partition, 
of which we have spoken on p. 349, and the annular film is very 
easy to reproduce by means of a soap-bubble in the bottom of 
a cylindrical dish or beaker. The next partition is a periclinal 
one, concentric with the outer surface of the young antheridium ; 
and this in turn is followed by a concave partition which cuts off 
the apex of the original cell: but which becomes connected with 
the second, or periclinal partition in precisely the same annular 
fashion as the first partition did with the base of the little 
antheridium. The result is that, at this stage, we have four 
cell-cavities in the little antheridium: (1) a central cavity; 
(2) an annular space around the lower margin; (3) a narrow annular 
or cylindrical space around the sides of the antheridium; and 
(4) a small terminal or apical cell. It is evident that the tendency, 
in the next place, will be to subdivide the flattened external cells 
by means of anticlinal partitions, and so to convert the whole 
structure into a single layer of epidermal cells, surrounding a 
central cell within which, in course of time, the antherozoids are 
developed. 


The foregoing account deals only with a few elementary pheno- 
mena, and may seem to fall far short of an attempt to deal in general 
with “the forms of tissues.”” But it is the principle involved, 
and not its ultimate and very complex results, that we can alone 


410 THE FORMS OF TISSUES ETC. [CH. VIII 


attempt to grapple with. The stock-in-trade of mathematical 
physics, in all the subjects with which that science deals, is for the 
most part made up of simple, or simplified, cases of phenomena 
which in their actual and concrete manifestations are usually too 
complex for mathematical analysis; and when we attempt to 
apply its methods to our biological and histological phenomena, 
in a preliminary and elementary way, we need not wonder if we 
be limited to illustrations which are obviously of a simple kind, 
and which cover but a small part of the phenomena with which 
the histologist has become familiar. But it is only relatively that 
these phenomena to which we have found the method applicable 
are to be deemed simple and few. They go already far beyond 
the simplest phenomena of all, such as we see in the dividing 
Protococcus, and in the first stages, two-celled or four-celled, of 
the segmenting egg. They carry us into stages where the cells 
are already numerous, and where the whole conformation has 
become by no means easy to depict or visualise, without the help 
and guidance which the phenomena of surface-tension, the laws 
of equilibrium and the principle of minimal areas are at hand 
to supply. And so far as we have gone, and so far as we can 
discern, we see no sign of the guiding principles failing us, or of 
the simple laws pas to hold good. 


CHAPTER IX 
ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS 


The deposition of inorganic material in the living body, usually 
in the form of calcium salts or of silica, is a very common and 
wide-spread phenomenon. It begins in simple ways, by the 
appearance of small isolated particles, crystalline or non- 
crystalline, whose form has little relation or sometimes none to 
the structure of the organism; it culminates in the complex 
skeletons of the vertebrate animals, in the massive skeletons of 
the corals, or in the polished, sculptured and mathematically 
regular molluscan shells. Hven among many very simple organ- 
isms, such as the Diatoms, the Radiolarians, the Foraminifera, 
or the Sponges, the skeleton displays extraordinary variety and 
beauty, whether by reason of the intrinsic form of its elementary 
constituents or the geometric symmetry with which these are 
arranged and interconnected. 

With regard to the form of these various structures (and this 
is all that immediately concerns us here), it is plain that we have 
to do with two distinct problems, which however, though 
theoretically distinct, may merge with one another. For the 
form of the spicule or other skeletal element may depend simply 
upon its chemical nature, as for instance, to take a simple but 
not the only case, when the form is purely crystalline; or the 
inorganic solid material may be laid down in conformity with the 
shapes assumed by the cells, tissues or organs, and so be, as it 
were, moulded to the shape of the living organism; and again, 
there may well be intermediate stages in which both phenomena 
may be simultaneously recognised, the molecular forces playing 
their part in conjunction with, and under the restraint of, the 
other forces inherent in the system. 


412 ON CONCRETIONS, SPICULES, [cH. 


So far as the problem is a purely chemical one, we must deal 
with it very briefly indeed; and all the more because special 
investigations regarding it have as yet been few, and even the 
main facts of the case are very imperfectly known. This at least 
is evident, that the whole series of phenomena with which we are 
about to deal go deep into the subject of colloid chemistry, and 
especially with that branch of the science which deals with the 
properties of colloids in connection with capillary or surface 
phenomena. It is to the special student of colloid chemistry that 
we must ultimately and chiefly look for the elucidation of our 
problem*. 

In the first and simplest part of our subject, the essential 
problem is the problem of crystallisation in presence of colloids. 
In the cells of plants, true crystals are found in comparative 
abundance, and they consist, in the great majority of cases, of 
calcium oxalate. Inthe stem and root of the rhubarb, for instance, 
in the leaf-stalk of Begonia, and in countless other cases, sometimes 
within the cell, sometimes in the substance of the cell-wall, we 
find large and well-formed crystals of this salt; their varieties of 
form, which are extremely numerous, are simply the crystalline 
forms proper to the salt itself, and belong to the two systems, 
cubic and monoclinic, in one or other of which, according to 
the amount of water of crystallisation, this salt is known to 
crystallise. When calcium oxalate crystallises according to the 
latter system (as it does when its molecule is combined with two 
molecules of water of crystallisation), the microscopic crystals 
have the form of fine needles, or “raphides,”’ such as are very 
common in plants; and it has been found that these are artificially 
produced when the salt is crystallised out in presence of glucose 
or of dextrin7. 

Calcium carbonate, on the other hand, when it occurs in plant- 
cells (as it does abundantly, for instance in the “cystoliths” of the 
Urticaceae and Acanthaceae, and in great quantities in Melobesia 

* There is much information regarding the chemical composition and minera- 
logical structure of shells and other organic products in H. C. Sorby’s Presidential 
Address to the Geological Society (Proc. Geol. Soc. 1879, pp. 56-93); but Sorby 
failed to recognise that association with “organic”? matter, or with colloid matter 


whether living or dead, introduced a new series of purely physical phenomena. 
+ Vesque, Ann. des Sc. Nat. (Bot.) (5), X1x, p. 310, 1874. 


1x] AND SPICULAR SKELETONS 413 


and the other calcareous or “stony” algae), appears in the form 
of fine rounded granules, whose inherent crystalline structure 
is not outwardly visible, but is only revealed (like that of a 
molluscan shell) under polarised light. Among animals, a skeleton 
of carbonate of lime occurs under a multitude of forms, of which 
we need only mention now a very few of the most conspicuous. 
The spicules of the calcareous sponges are triradiate, occasionally 
quadriradiate, bodies, with pointed rays, not crystalline in outward 
form but with a definitely crystalline internal structure. We shall 


Fig. 194. Alcyonarian spicules: Siphonogorgia and Anthogorgia. (After Studer.) 


return again to these, and find for them what would seem to be 
a satisfactory explanation of their form. Among the Alcyonarian 
zoophytes we have a great variety of spicules*, which are some- 
times straight and slender rods, sometimes flattened and more or 
less striated plates, and still more often rounded or branched 
concretions with rough or knobby surfaces (Figs. 194, 200). A 
third type, presented by several very different things, such as 
a pearl, or the ear-bone of a bony fish, consists of a more or less 


* Cf. Kélliker, [cones Histiologicae, 1864, pp. 119, etc. 


414 ON CONCRETIONS, SPICULES, [CH. 


rounded body, sometimes spherical, sometimes flattened, in which 
the calcareous matter is laid down in concentric zones, denser 
and clearer layers alternating with one another. In the develop- 
ment of the molluscan shell and in the calcification of a bird’s 
egg or the shell of a crab, for instance, spheroidal bodies with 
similar concentric striation make their appearance; but instead of 
remaining separate they become crowded together, and as they 
coalesce they combine to form a pattern of hexagons. In some 
cases, the carbonate of lime on being dissolved away by acid 
leaves behind it a certain small amount of organic residue; in 
most cases other salts, such as phosphates of lime, ammonia or 
magnesia are present in small quantities; and in most cases if 
not all the developing spicule or concretion is somehow or other 
so associated with living cells that we are apt to take it for granted 
that it owes its peculiarities of form to the constructive or plastic 
agency of these. 

The appearance of direct association with living cells, however, 
is apt to be fallacious; for the actual precipitation takes place, 
as a rule, not in actively living, but in dead or at least inactive 
tissue*: that is to say in the “formed material” or matrix which 
(as for instance in cartilage) accumulates round the living cells, 
in the interspaces between these latter, or at least, as often happens, 
in connection with the cell-wall or cell-membrane rather than 
within the substance of the protoplasm itself. We need not go 
the length of asserting that this is a rule without exception; but, 
so far as it goes, it is of great importance and to its consideration - 
we shall presently return 7. 

Cognate with this is the fact that it is known, at least in some ~ 
cases, that the organism can go on living and multiplying with 
apparently unimpaired health, when stinted or even wholly 
deprived of the material of which it is wont to make its spicules 


* Tn an interesting paper by Irvine and Sims Woodhead on the “Secretion of 
Carbonate of Lime by Animals” (Proc. R. S. E. xvi, 1889, p. 351) it is asserted 
that “‘lime salts, of whatever form, are deposited only in vitally inactive tissue.” 

+ The tube of Teredo shews no trace of organic matter, but consists of irregular 
prismatic crystals: the whole structure ‘ being identical with that of small veins 
of calcite, such as are seen in thin sections of rocks”’ (Sorby, Proc. Geol. Soc. 1879, 
p. 58). This, then, would seem to be a somewhat exceptional case of a shell laid 
down completely outside of the animal’s external layer of organic or colloid sub- 
stance. 


Ix] AND SPICULAR SKELETONS 415 


or its shell. Thus, Pouchet and Chabry* have shown that the 
eggs of sea-urchins reared in lime-free water develop in apparent 
’ health, into larvae entirely destitute of the usual skeleton of 
calcareous rods, and in which, accordingly, the long arms of the 
Pluteus larva, which the rods support and distend, are entirely 
suppressed. And again, when Foraminifera are kept for genera- 
tions in water from which they gradually exhaust the lime, their 
shells grow hyaline and transparent, and seem to consist only of 
chitinous material. On the other hand, in the presence of excess 
of lime, the shells become much altered, strengthened with various 
“ornaments,” and assuming characters described as proper to 
other varieties and even species f. 

The crucial experiment, then, is to attempt the formation of 
similar structures or forms, apart from the living organism: but, 
however feasible the attempt may be in theory, we shall be prepared 
from the first to encounter difficulties, and to realise that, though 
the actions involved may be wholly within the range of chemistry 
and physics, yet the actual conditions of the case may be so 
complex, subtle and delicate, that only now and then, and in the 
simplest of cases, shall we find ourselves in a position to imitate 
them completely and successfully. Such an investigation is only 
part of that much wider field of enquiry through which Stephane 
Leduc and many other workerst have sought to produce, by 
synthetic means, forms similar to those of living things; but it 
is a well-defined and circumscribed part of that wider investigation. 
When by chemical or physical experiment we obtain configurations 
similar, for instance, to the phenomena of nuclear division, or 
_ conformations similar to a pattern of hexagonal cells, or a group 
of vesicles which resemble some particular tissue or cell-aggregate, 
we indeed prove what it is the main object of this book to illustrate, 
namely, that the physical forces are capable of producing particular 
organic forms. But it is by no means always that we can feel 
perfectly assured that the physical forces which we deal with in 
our experiment are identical with, and not merely analogous to, 


* CR. Soc. Biol. Paris (9), 1, pp. 17-20, 1889; C. R. Ac. Sc. cvim. pp. 196-8, 
1889. 

7 Cf. Heron-Allen, Phil. Trans. (B), vol. ccvi1, p. 262, 1915 

t See Leduc, Mechanism of Life (1911), ch. x, for copious references to other 
works on the artificial production of “organic” forms. 


416 ON CONCRETIONS, SPICULES, ETC. [CH. 


the physical forces which, at work in nature, are bringing about 
the result which we have succeeded in imitating. In the present 
case, however, our enquiry is restricted and apparently simplified ; 
we are seeking in the first instance to obtain by purely chemical 
means a purely chemical result, and there is little room for 
ambiguity in our interpretation of the experiment. ' 


When we find ourselves investigating the forms assumed by 
chemical compounds under the peculiar circumstances of associa- 
tion with a living body, and when we find these forms to be 
characteristic or recognisable, and somehow different from those 
which, under other circumstances, the same substance is wont 
to assume, an analogy presents itself to our minds, captivating 
though perhaps somewhat remote, between this subject of ours 
and certain synthetic problems of the organic chemist. There is 
doubtless an essential difference, as well as a difference of scale, 
between the visible form of a spicule or concretion and the hypo- 
thetical form of an individual molecule; but molecular form is 
a very important concept; and the chemist has not only succeeded, 
since the days of Wohler, in synthesising many substances which 
are characteristically associated with living matter, but his task 
has included the attempt to account for the molecular forms of 
certain “asymmetric” substances, glucose, malic acid and many 
more, as they occur in nature. These are bodies which, when 
artificially synthesised, have no optical activity, but which, as we 
actually find them in organisms, turn (when 7n solution) the plane 
of polarised light in one direction or the other; thus dextro- 
glucose and laevomalic acid are common products of plant 
metabolism; but dextromalic acid and laevo-glucose do not occur 
in nature at all. The optical activity of these bodies depends, 
as Pasteur shewed more than fifty years ago*, upon the form, 
right-handed or left-handed, of their molecules, which molecular 
asymmetry further gives rise to a corresponding right or left- 
handedness (or enantiomorphism) in the crystalline aggregates. 
It is a distinct problem in organic or* physiological chemistry, 

* Lectures on the Molecular Asymmetry of Natural Organic Compounds, 
Chemical Soc. of Paris, 1860, and also in Ostwald’s Klassiker d. ex. Wiss. No. 28, 


and in Alembic Club Reprints, No. 14, Edinburgh, 1897; cf. Richardson, G. M., 
Foundations of Stereochemistry, N. Y. 1901. 


1x] OF MOLECULAR ASYMMETRY 417 


and by no means without its interest for the morphologist, to 
discover how it is that nature, for each particular substance, 
habitually builds up, or at least selects. its molecules in a one- 
sided fashion, right-handed or left-handed as the case may be. 
Tt will serve us no better to assert that this phenomenon has its 
origin in “fortuity,” than to repeat the Abbé Galiani’s saying, 
“les dés de la nature sont pipés.” 

The problem is not so closely related to our immediate subject 
that we need discuss it at length; but at the same time it has its 
clear relation to the general question of form in relation to vital 
phenomena, and moreover it has acquired interest as a theme 
of long-continued discussion and new importance from some 
comparatively recent discoveries. 

According to Pasteur, there lay in the molecular asymmetry 
of the natural bodies and the symmetry of the artificial products, 
one of the most deep-seated differences between vital and non- 
vital phenomena: he went further, and declared that “this was 
perhaps the only well-marked line of demarcation that can at 
present [1860] be drawn between the chemistry of dead and of 
living matter.” Nearly forty years afterwards the same theme 
was pursued and elaborated by Japp in a celebrated lecture™, 
and the distinction still has its weight, I believe, in the minds of 
many if not most chemists. 

“We arrive at the conclusion,” said Professor Japp, “that the 
production of single asymmetric compounds, or their isolation 
from the mixture of their enantiomorphs, is, as Pasteur firmly 
held, the prerogative of life. Only the living organism, or the 
living intelligence with its conception of asymmetry, can produce 


2 


this result. Only asymmetry can beget asymmetry.” In these 
last words (which, so far as the chemist and the biologist are 
concerned, we may acknowledge to be perfectly truet) lies the 


* Japp, Stereometry and Vitalism, Brit. Ass. Rep. (Bristol), p. 813, 1898; 
ef. also a voluminous discussion in Natwre, 1898-9. 

+ They represent the general theorem of which particular cases are found, for 
instance, in the asymmetry of the ferments (or enzymes) which act upon 
asymmetrical bodies, the one fitting the other, according to Emil Fischer’s well- 
known phrase, as lock and key. Cf. his Bedeutung der Stereochemie fiir die 
Physiologie, Z. f. physiol. Chemie, v, p. 60, 1899, and various papers in the Ber. 
d. d. chem. Ges. from 1894. 


T. GC. 27 


418 ON CONCRETIONS, SPICULES, ETC. [CH. 


crux of the difficulty; for they at once bid us enquire whether in 
nature, external to and antecedent to life, there be not some 
asymmetry to which we may refer the further propagation or 
“begetting’” of the new asymmetries: or whether in default 
thereof, we be rigorously confined to the conclusion, from which 
Japp “saw no escape,” that “at the moment when life first arose, 
a directive force came into play,—a force of precisely the same 
character as that which enables the intelligent operator, by the 
exercise of his will, to select one crystallised enantiomorph and 
reject its asymmetric opposite *.” 

Observe that it is only the first beginnings of chemical 
asymmetry that we need to discover; for when asymmetry is once 
manifested, it is not disputed that it will continue “to beget 
asymmetry.” A plausible suggestion is now at hand, which if it 
be confirmed and extended will supply or at least sufficiently 
illustrate the kind of explanation which is required f. 

We know in the first place that in cases where ordinary non- 
polarised light acts upon a chemical substance, the amount of 
chemical action is proportionate to the amount of light absorbed. 
We know in the second place, in certain cases, that hght circularly 
polarised is absorbed in different amounts by the right-handed or 
left-handed varieties, as the case may be, of an asymmetric 
substance. And thirdly, we know that a portion of the lght 
which comes to us from the sun is already plane-polarised light, 
which becomes in part circularly polarised, by reflection (according 
to Jamin) at the surface of the sea, and then rotated in a 
particular direction under the influence of terrestrial magnetism. 
We only require to be assured that the relation between ab- 
sorption of light and chemical activity will continue to hold 
good in the case of circularly polarised light; that is to say 


* In accordance with Emil Fischer’s conception of “asymmetric synthesis,” 


it is now held to be more likely that the process is synthetic than analytic: more 
likely, that is to say, that the plant builds up from the first one asymmetric body 
to the exclusion of the other, than that it “selects” or “picks out” (as Japp sup- 
posed) the right-handed or the left-handed molecules from an original, optically 
inactive, mixture of the two; cf. A. McKenzie, Studies in Asymmetric Synthesis, 
Journ. Chem. Soc. (Trans.), LXxxv, p. 1249, 1904. 

+ See for a fuller discussion, Hans Przibram, Vitalitdt, 1913, Kap. iv, Stoff- 
wechsel (Assimilation und Katalyse). 

t Cf. Cotton, Ann. de Chim. et de Phys. (7), vit, pp. 347-432 (cf. p. 373), 1896. 


Ix] OF MOLECULAR, ASYMMETRY 419 


that the formation of some new substance or other, under the 
influence of light so polarised, will proceed asymmetrically in 
consonance with the asymmetry of the light itself; or conversely, 
that the asymmetrically polarised light will tend to more rapid 
decomposition of those molecules by which it is chiefly absorbed. 
This latter proof is now said to be furnished by Byk*, who asserts 
that certain tartrates become unsymmetrical under the continued 
influence of the asymmetric rays. Here then we séem to have 
an example, of a particular kind and in a particular instance, an 
example limited but yet crucial (if confirmed), of an asymmetric 
force, non-vital in its origin, which might conceivably be the 
starting-point of that asymmetry which is characteristic of so 
many organic products. 

The mysteries of organic chemistry are great, and the differences 
between its processes or reactions as they are carried out in the 
organism and in the laboratory are many}. The actions, catalytic 
and other, which go on in the living cell are of extraordinary 
complexity. But the contention that they are different in kind 
from what we term ordinary chemical operations, or that in the 
production of single asymmetric compounds there is actually to 
be witnessed, as Pasteur maintained, a “prerogative of life,” 
would seem to be no longer safely tenable. And furthermore, it 
behoves us to remember that, even though failure continued to 
attend all artificial attempts to originate the asymmetric or 
optically active compounds which organic nature produces in 
abundance, this would only prove that a certain physical force, or 
mode of physical action, is at work among living things though 
unknown elsewhere. It is a mode of action which we can easily 
imagine, though the actual mechanism we cannot set agoing when 
we please. And it follows that such a difference between living 
matter and dead would carry us but a little way, for it would still . 
be confined strictly to the physical or mechanical plane. 

Our historic interest in the whole question is increased by the 

* Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch Zirkular- 
polarisiertes Licht, ein Beitrag zur primaren Entstehung optisch-activer Substanzen, 
Zeitsch. f. physikal. Chemie, xurx, p. 641, 1904. It must be admitted that further 
positive evidence on these lines is still awanting. 

+ Cf. (int. al.) Emil Fischer, Untersuchungen iiber Aminosduren, Proteine, ete. 
Berlin, 1906. 

27—2 


420 ON CONCRETIONS, SPICULES, ETC. © [ox-: 


fact, or the great probability, that “the tenacity with which 
Pasteur fought against the doctrine of spontaneous generation was 
not unconnected with his belief that chemical compounds of one- 
sided symmetry could not arise save under the influence of life *.” 
But the question whether spontaneous generation be a fact or not 
does not depend upon theoretical considerations; our negative 
response is based, and is so far soundly based, on repeated failures 
to demonstrate its occurrence. Many a great law of physical 
science, not excepting gravitation itself, has no higher claim on 
our acceptance. 


Let us return then, after this digression, to the general subject 
of the forms assumed by certain chemical bodies when deposited 
or precipitated within the organism, and to the question of how 
far these forms may be artificially imitated or theoretically 
explained. 

Mr George Rainey, of St Bartholomew’s Hospital (to whom 
we have already referred), and Professor P. Harting, of Utrecht, 
were the first to deal with this specific problem. Mr Rainey 
published, between 1857 and 1861, a series of valuable and 
thoughtful papers to shew that shell and bone and certain other 
organic structures were formed “by a process of molecular 
coalescence; demonstrable in certain artificially-formed products fT.” 
Professor Harting, after thirty vears of experimental work, 

ublished in 1872 a paper, which has become classical, entitled 
ib pap 
Recherches de Morphologie Synthétique, sur la production artificielle 
de quelques formations calcaires organiques; his aim was to pave 
the way for a “morphologie synthétique,’ as Wohler had laid the 
foundations of a “‘chimie synthétique,” by his classical discovery 
forty years before. 

* Japp, J. c. p. 828. 

+ Rainey, G., On the Elementary Formation of the Skeletons of Animals, and 
other Hard Structures formed in connection with Living Tissue, Brit. For. Med. 
Ch. Rev. xx, pp. 451476, 1857; published separately with additions, 8vo. London, 
1858. For other papers by Rainey on kindred subjects see Q. J. M. S. vi (Tr. 
Microsc. Soc.), pp. 41-50, 1858, vu, pp. 212-225, 1859, vin, pp. 1-10, 1860, 
I (n. s.), pp. 23-32, 1861. Cf. also Ord, W. M., On Molecular Coalescence, and on 
the influence exercised by Colloids upon the Forms of Inorganic Matter, Q. J. M. S. 
XI, pp. 219-239, 1872; and also the early but still interesting observations of 


Mr Charles Hatchett, Chemical Experiments on Zoophytes; with some observa- 
tions on the component parts of Membrane, Phil. Trans. 1800, pp. 327-402. 


Ix] HARTING’S MORPHOLOGIE SYNTHETIQUE 421 


Rainey, and Harting used similar methods, and these were 
such as many other workers have continued to employ,—partly 
with the direct object of explaining the genesis of organic forms 
and partly as an integral part of what is now known as Colloid 
Chemistry. The whole gist of the method was to bring some soluble 
salt of lime, such as the chloride or nitrate, into solution within a 
colloid medium, such as gum, gelatine or albumin; and then to 
precipitate it out in the form of some insoluble compound, such 
as the carbonate or oxalate. Harting found that, when he added 
a little sodium or potassium carbonate to a concentrated solution 
of calcium chloride in albumin, he got at first a gelatinous mass, 
or “colloid precipitate”: which slowly transformed by the 


Fig. 195. Calcospherites. orconcretions Fig. 196. A single calco- 


of calcium carbonate, deposited in spherite, with central 

white of egg. (After Harting.) “nucleus,” and _ striated, 
iridescent border. (After 
Harting.) 


appearance of tiny microscopic particles, at first motionless, but 
afterwards as they grew larger shewing the typical Brownian 
movement. So far, very much the same phenomena were wit- 
nessed whether the solution were albuminous or not, and similar 
appearances indeed had been witnessed and recorded by Gustav 
Rose, so far back as 1837*; but in the later stages the presence 
of albuminoid matter made a great difference. Now, after a few 
days, the calcium carbonate was seen to be deposited in the form 
of large rounded concretions, with a more or less distinct central 
nucleus, and with a surrounding structure at once radiate and 


* Cf. Quincke, Ueber unsichtbare Fliissigkeitsschichten, Ann. der Physik, 1902. 


422 ON CONCRETIONS, SPICULES, ETC. [CH. 


concentric; the presence of concentric zones or lamellae, alter- 

nately dark and clear, was especially characteristic. These 

round “calcospherites’’ shewed a tendency to aggregate together 
t 


Fig. 197. Later stages in the same experiment, 


in layers, and then to assume polyhedral, or often regularly 
hexagonal, outlines. In this latter condition they closely resemble 


Fig. 198, A. Section of shell of Mya; B. Section of hinge- 
tooth of do. (After Carperiter.) 


the early stages of calcification in a molluscan (Fig. 198), or still 
more in a crustacean shell*; while in their isolated condition 


* See for instance other excellent illustrations in Carpenter’s article “Shell,” in 
Todd’s Cyclopedia, vol. tv. pp. 556-571, 1847-49. According to Carpenter, the 
shells of the mollusca (and also of the crustacea) are “essentially composed of 
cells, consolidated by a deposit of carbonate of lime in their interior.” That is 
to say, Carpenter supposed that the spherulites, or calcospherites of Harting, were, 
to begin with, just so many living protoplasmic cells. Soon afterwards .however, 


Ix] ON SPHERULITES OR CALCOSPHERITES 423 


they very closely resemble the little calcareous bodies in the 
tissues of a trematode or a cestode worm, or in the oesophageal 
glands of an earthworm*. 

When the albumin was somewhat scanty, or when it was mixed 
with gelatine, and especially when a little phosphate of lime was 


Fig. 199. Large irregular calcareous concretions, or spicules, deposited in a piece 
of dead cartilage, in presence of calcium phosphate. (After Harting.) 


Huxley pointed out that the mode of formation. while at first sight ‘irresistibly 
suggesting a cellular structure,...is in reality nothing of the kind,” but *‘is simply 
the result of the concretionary manner in which the calcareous matter is deposited ” ; 
ibid. art. ““Tegumentary Organs,” vol. v, p. 487, 1859. . Quekett (Lectures on 
Histology, vol. u, p. 393, 1854, and Q. J. M. S. x1, pp. 95-104, 1863) supported 
Carpenter; but Williamson (Histological Features in the Shells of the Crustacea, 
Q. J. M. S. vi, pp. 35-47, 1860) amply confirmed Huxley’s view, which in the 
end Carpenter himself adopted (The Microscope, 1862, p. 604). A like controversy 
arose later in regard to corals. Mrs Gordon (M. M. Ogilvie) asserted that the coral 
was built up “of successive layers of calcified cells, which hang together at first by 
their cell-walls, and ultimately, as crystalline changes continue, form the individual 
laminae of the skeletal structures” (Phil. Trans. cLXxxxvit, p. 102, 1896): whereas 
v. Koch had figured the coral as formed out of a mass of “ Kalkconcremente”’ 
or “erystalline spheroids,” laid down outside the ectoderm, and precisely similar 
both in their early rounded and later polygonal stages (though von Koch was not 
aware of the fact) to the calecospherites of Harting (Entw. d. Kalkskelettes von 
Asteroides, Mitth. Zool. St. Neapel, 111, pp. 284-290, pl. xx, 1882). Lastly Duerden 
shewed that external to, and apparently secreted by the ectoderm lies a homo- 
geneous organic matrix or membrane, “in which the minute calcareous crystals 
forming the skeleton are laid down” (The Coral Siderastraea radians, ete., Carnegie 
Inst. Washington, 1904, p. 34). Cf. also M. M. Ogilvie-Gordon, Q. J. M.S. xurx, 
p- 203, 1905, ete. 
* Cf. Claparéde, Z. f. w. Z. x1x, p. 604, 1869. 


424 ON CONCRETIONS, SPICULES, ETC. [CH. 


added to the mixture, the spheroidal globules tended to become 
rough, by an outgrowth of spinous or digitiform projections; and 
in some cases, but not without the presence of the phosphate, the 
result was an irregularly shaped knobby spicule, precisely similar 
to those which are characteristic of the Alcyonaria*. 


The rough spicules of the Aleyonaria are extraordinarily variable in shape 
and size, as, looking at them from the chemist’s or the physicist’s point of 
view, we should expect them to be. Partly upon the form of these spicules, 
and partly on the general form or mode of branching of the entire colony of 


Fig. 200. Additional illustrations of Alcyonarian spicules: Hunicea. (After 
Studer.) 


polypes, a vast number of separate “species” have been based by systematic 
zoologists. But it is now admitted that even in specimens of a single species, 
from one and the same locality, the spicules may vary immensely in shape 
and size: and Professor Hickson declares (in a paper published while these 
sheets are passing through the press) that after many years of laborious work 
in striving to determine species of these animal colonies, he feels “‘ quite con- 
vinced that we have been engaged in a more or less fruitless task}. 

The formation of a tooth has very lately been shown to be a phenomenon 
of the same order. That is to say, * calcification in both dentine and enamel 

* Spicules extremely like those of the Aleyonaria occur also in a few sponges; 
ef. (e.g.), Vaughan Jennings, Journ. Linn. Soc. xxut, p. 531, pl. 13, fig. 8, 1891. 

+ Mem. Manchester Lit. and Phil. Soc. ux, p. 11, 1916. 


Ix] ON SPHERULITES OR CALCOSPHERITES 425 


is in great part a physical phenomenon; the actual deposit in both tissues 
occurs in the form of calcospherites, and the process in mammalian tissue 
is identical in every point with the same process occurring in lower organisms*.” 
The ossification of bone, we may be sure, is in the same sense and to the same 
extent a physical phenomenon. 


The typical structure of a calcospherite is no other than that 
of a pearl, nor does it differ essentially from that of the otolith 
of a molluse or of a bony fish. (The otoliths, by the way, of the 
elasmobranch fishes, like those of reptiles and birds, are not 
developed after this fashion, but are true crystals of calc-spar.) 

Throughout these phenomena, the effect of surface-tension is 
manifest. It is by surface-tension that ultra-microscopic particles 
are brought together in the first floccular precipitate or coagulum ; 


Pp 
4 


Fig. 201, A “crust” of close-packed Y 

sleatcous gncreons presntled pig, Aggregated alo 

solution. (After Harting.) spherites. (After Harting.) 
by the same agency, the coarser particles are in turn agglutinated 
into visible lumps; and the form of the calcospherites, whether 
it be that of the solitary spheres or that assumed in various stages 
of aggregation (e.g. Fig. 202) f, is likewise due to the same agency. 
From the point of view of colloid chemistry the whole phe- 
nomenon is very important and significant; and not the least 
significant part is this tendency of the solidified deposits to assume 
the form of “spherulites,” and other rounded contours. In the 
phraseology of that science, we are dealing with a two-phase 
system, which finally consists of solid particles in suspension in 
a liquid (the former being styled the disperse phase, the latter the 


* Mummery, J. H., On Calcification in Enamel and Dentine, Phil. Trans. coy 
(B), pp. 95-111, 1914. 

+ The artificial concretion represented in Fig. 202 is identical in appearance 
with the concretions found in the kidney of Nautilus, as figured by Willey (Zoological 
Results, p. \xxvi, Fig. 2, 1902). 


5 


426 ON CONCRETIONS, SPICULES, ETC. [CH. 


dispersion medium). In accordance with a rule first recognised 
by Ostwald*, when a substance begins to separate out from a 
solution, so making its appearance as a new phase, it always 
makes its appearance first as a liquid}. Here is a case in point. 
The minute quantities of material, on their way from a state of 
solution to a state of “suspension,” pass through a lquid*to a 
solid form; and their temporary sojourn in the former leaves its 
impress in the rounded contours which surface-tension brought 
about while the little aggregate was still labile or fluid: while 
coincidently with this surface-tension effect upon the surface, 
crystallisation tended to take place throughout the little liquid 


mass, or in such portion of it as had not yet consolidated and 
crystallised. 


AMM UI SN 


Fig. 203. (After Harting.) 


Where we have simple aggregates of two or three calcospherites, 
the resulting figure is precisely that of so many contiguous soap- 
bubbles. In other cases, composite forms result which are not 
so easily explained, but which, if we could only account for them, 
would be of very great interest to the biologist. For instance, 
when smaller calcospheres seem, as it were, to invade the substance 
of a larger one, we get curious conformations which in the closest 
possible way resemble the outlines of certain of the Diatoms 
(Fig. 203). Another very curious formation, which Harting calls 
a “conostat,” is of frequent occurrence, and in it we see at least 
a suggestion of analogy with the configuration which, in a proto- 
plasmic structure, we have spoken of as a “collar-cell.” The 

* Cf. Taylor’s Chemistry of Colloids, p. 18, etc., 1915. 

+ This rule, undreamed of by Errera, supports and justifies the cardinal 
assumption (of which we have had so much to say in discussing the forms of cells 


and tissues) that the incipient cell-wall behaves as, and indeed actually is, a liquid 
film (cf. p. 306). 


Ix] OF LIESEGANG’S RINGS 427 


conostats, which are formed in the surface layer of the solution, 
consist of a portion of a spheroidal calcospherite, whose upper 
part is continued into a thin spheroidal collar, of somewhat larger 
radius than the solid sphere; but the precise manner in which 
the collar is formed, possibly around a bubble of gas, possibly 
about a vortex-like diffusion-current* is not obvious. 


Among these various phenomena, the concentric striation 
observed in the calcospherite has acquired a special interest and 
importance. It is part of a phenomenon now widely known, and 
recognised as an important factor in colloid chemistry, under the 
name of “Liesegang’s Ringst.” 


Fig. 204. Conostats. (After Harting.) 


If we dissolve, for instance, a little bickromate of potash in 
gelatine, pour it on to a glass plate, and after it is set place upon 
it a drop of silver nitrate solution, there appears in the course 
of a few hours the phenomenon of Liesegang’s rings. At first the 
silver forms a central patch of abundant reddish brown chromate 
precipitate; but around this, as the silver nitrate diffuses slowly 
through the gelatine, the precipitate no longer comes down in 
a continuous, uniform laver, but forms a series of zones, beautifully 
regular, which alternate with clear interspaces of jelly, and which 
stand farther and farther apart, in logarithmic ratio, as they 
recede from the centre. For a discussion of the ravson d’étre of 

* Cf. p. 254. 

+ Cf. Harting, op. cit., pp. 22, 50: ‘‘J’avais cru d’abord que ces couches 
concentriques étaient produites par l’alternance de la chaleur ou de la lumieére, 
pendant le jour et la nuit. Mais lexpérience, expressément instituée pour 
examiner cette question, y a répondu négativement.” 


{ Liesegang, R. E., Ueber die Schichtungen bei Diffusionen, Leipzig, 1907, and 
other earlier papers. 


428 ON CONCRETIONS, SPICULES, ETC. [CH. 


this phenomenon, still somewhat problematic, the student must» 
consult the text-books of physical and colloid chemistry*. 

But, speaking very generally, we may say the appearance of 
Liesegang’ s rings is but a particular and striking case of a more 
general phenomenon, namely es influence on crystallisation of 
the presence of foreign bodies or “impurities,” represented in this 
case by the ° eel or colloid matrix}. Faraday shewed long ago 
that to the presence of slight impurities might be ascribed the 
banded structure of ice, of banded quartz or agate, onyx, etc. ; 
and Quincke and Tomlinson have added to our scanty knowledge 
of the same phenomenon tf. 


Fig. 205. Liesegang’s Rings. (After Leduc.) 


Besides the tendency to rhythmic action, as manifested in 
Liesegang’s rings, the association of colloid matter with a crystal- 
loid in solution may lead to other well-marked effects. These, 
according to Professor J. H. Bowman§, may be grouped somewhat 
as follows: (1) total prevention of crystallisation ; (2) suppression of 
certain of the lines of crystalline growth; (3) extension of the crystal 
to abnormal proportions, with a tendency for it to become a com- 
pound crystal; (4) a curving or gyrating of the crystal or its parts. 

* Cf. Taylor’s Chemistry of Colloids, pp. 146-148, 1915. 

+ Cfé. 8. C. Bradford, The Liesegang Phenonemon and Concretionary Structure 
in Rocks, Nature, xcvu, p. 80, 1916; ef. Sci. Progress, x, p. 369, 1916. 

{ Cf. Faraday, On Ice of Irregular Fusibility, Phil. Trans., 1858, p. 228; 
Researches in Chemistry, etc., 1859, p. 374; Tyndall, Forms of Water, p. 178, 
1872; Tomlinson, C., On some effects of small Quantities of Foreign Matter on 
Crystallisation, Phil. Mag. (5) xxxt, p. 393, 1891, and other papers. 

§ A Study in Crystallisation, J. of Soc. of Chem. Industry, xxv, p. 143, 1906. 


Ix] OF LIESEGANG’S RINGS 429 


For instance, it would seem that, if the supply of:material to 
the growing crystal be not forthcoming in sufficient quantity (as 
may well happen in a colloid medium, for lack of convection- 
currents), then growth will follow only the strongest lines of 
crystallising force, and will be suppressed or partially suppressed 
along other axes. The crystal will have a tendency to become 
filiform, or “fibrous”; and the raphides of our plant-cells are 
a casein point. Again, the long slender crystal so formed, pushing 
its way into new material, may initiate a new centre of crystallisa- 
tion: we get the phenomenon known as a “relay,” along the 


Fig. 206. Relay-crystals of common salt. (After Bowman.) 


principal lines of force, and sometimes along subordinate axes as 
well. This phenomenon is illustrated in the accompanying figure 
of crystallisation -in a colloid medium of common salt; and 
it may possibly be that we have here an explanation, or 


part of an explanation, of the i) 
* = 


compound siliceous spicules of | IN 
* cS 


the Hexactinellid sponges. 
Lastly, when the crystallising 
force is nearly equalled by 
the resistance of the viscous 
medium, the crystal takes the 
line of least resistance, with’ 
very various results. One of 
these results would seem to be 
a gyratory course, giving to Fig. 207. Wheel-like crystals in a 
the crystal a curious wheel-hke SOI. (Ae aes 
shape, as in Fig. 207; and other results are the feathery, fern-like 


430 ON CONCRETIONS, SPICULES, ETC. [CH. 


or arborescent shapes so frequently seen in microscopic crys- 
tallisation. 

To return to Liesegang’s rings, the typical appearance of 
concentric rings upon a gelatinous plate may be modified in 
various experimental ways. For instance, our gelatinous medium 
may be placed in a capillary tube immersed in a solution of the 
precipitating salt, and in this case we shall obtain a vertical 
succession of bands or zones regularly interspaced : the result being 
very closely comparable to the banded pigmentation which we see 
in the hair of a rabbit or a rat. In the ordinary plate preparation, 
the free surface of the gelatine is under different conditions to the 
lower layers and especially to the lowest layer in contact with 
the glass; and therefore it often happens that we obtain a double 
series of rings, one deep and the other superficial, which by 
occasional blending or interlacing, may produce a netted. pattern. 
In some cases, as when only the inner surface of our capillary 
tube is covered with a layer of gelatine, there is a tendency for 
the deposit to take place in a continuous spiral line, rather than 
in concentric and separate zones. By such means, according to 
Kiister* various forms of annular, spiral and reticulated thickenings 
in the vascular tissue of plants may be closely imitated; and he 
and certain other writers have of late been inclined to carry the 
same chemico-physical phenomenon a very long way, in the 
explanation of various banded, striped, and other rhythmically 
successional types of structure or pigmentation. For example, 
the striped pigmentation of the leaves in many plants (such as 
Eulalia japonica), the striped or clouded colourmg of many 
feathers or of a cat’s skin, the patterns of many fishes, such for 
instance as the brightly coloured tropical Chaetodonts and the like, 
are all regarded by him as so many instances of “ diffusion-figures” 
closely related to the typical Liesegang phenomenon. Gebhardt 
has made a particular study of the same subject in the case of 
‘insectst. He declares, for instance, that the banded wings of 
Papilio podalirius are precisely imitated in Liesegang’s experi- 
ments; that the finer markings on the wings of the Goatmoth 
(Cossus ligniperda) shew the double arrangement of larger and of 


* Ueber Zonenbildung in kolloidalen Medien. Jena, 1913. 
+ Verh. d. d. Zool. Gesellsch. p. 179, 1912. 


Ix] OF LIESEGANG’S RINGS 431 


smaller intermediate rhythms, likewise manifested in certain cases 
of the same kind; that the alternate banding of the antennae 
(for instance in Sesia spheciformis), a pigmentation not concurrent 
with the segmented structure of the antenna, is explicable in the 
same way; and that the “ocelli,’ for instance of the Emperor 
moth, are typical illustrations of the common concentric type. 
Darwin’s well-known disquisition* on the ocellar pattern of the 
feathers of the Argus Pheasant, as a result of sexual selection, 
will occur to the reader’s mind, in striking contrast to this or 
to any other direct physical explanationt. To turn from the dis- 
tribution of pigment to more deeply seated structural characters, 
Leduc has shewn how, for instance, the laminar structure of the 
cornea or the lens is again, apparently, a similar phenomenon. ~ 
In the lens of the fish’s eye, we have a very curious appearance, 
the consecutive lamellae being roughened or notched by close-set, 
interlocking sinuosities; and precisely the same appearance, save 
that it is not quite so regular, is presented in one of Kiister’s 
figures as the effect of precipitating a little sodium phosphate in 
a gelatinous medium. Biedermann has studied, from the same 
point of view, the structure and development of the molluscan 
shell, the problem which Rainey had first attacked more than 
fifty years beforet; and Liesegang himself has applied his results 
to the formation of pearls, and to the development of bone§. 

* Descent of Man, 1, pp. 132-153, 1871. 

+ As a matter of fact, the phenomena associated with the development of an 
“ocellus” are or may be of great complexity, inasmuch as they involve not only 
a graded distribution of pigment, but also, in “optical” coloration, a symmetrical 
distribution of structure or form. The subject therefore deserves very careful 
discussion, such as Bateson gives to it (Variation, chap. xii). This, by the way, 
is one of the very rare cases in which Bateson appears inclined to suggest a purely 
physical explanation of an organic phenomenon: “The suggestion is strong that 
the whole series of rings (in Morpho) may have been formed by some one central 
disturbance, somewhat as a series of concentric waves may be formed by the 
splash of a stone thrown into a pool, etc.” 

t Cf. also Sir D. Brewster, On optical properties of Mother of Pearl, Phil. Tans. 
1814, p. 397. 

§ Biedermann, W., Ueber die Bedeutung von Kristallisationsprozessen der 
Skelette wirbelloser Thiere, namentlich der Molluskenschalen, Z. f. allg. Physiol. 
I, p. 154, 1902; Ueber Bau und Entstehung der Molluskenschale, Jen. Zeitschr. 
Xxxvi, pp. 1-164, 1902. Cf. also Stemmann, Ueber Schale und Kalksteinbildungen, 


Ber. Naturf. Ges. Freiburg. Br. tv, 1889; Liesegang, Naturw. Wochenschr. p. 641, 
1910. : é 


432 ON CONCRETIONS, SPICULES, ETC. [CH. 


Among all the many cases where this phenomenon of Liese- 
gang’s comes to the naturalist’s aid in explanation of rhythmic or 
zonary configurations in organic forms, it has a special interest 
where the presence of concentric zones or rings appears, at 
first sight, as a sure and certain sign of periodicity of growth, 
depending on the seasons, and capable therefore of serving as 
a mark and record of the creature’s age. This is the case, for 
instance, with the scales, bones and otoliths of fishes; and a 
kindred phenomena in starch-grains has given rise, in ike manner, 
to the belief that they indicate a diurnal and nocturnal periodicity 
of activity and rest*. 

That this is actually the case in growing starch-grains is 
generally believed, on the authority of Meyer}; but while under 
certain circumstances a marked alternation of growing and resting 
periods may occur, and may leave its impress on the structure 
of the grain, there is now great reason to believe that, apart from 
such external influences, the internal phenomena of 
diffusion may, just as in the typical Liesegang 
experiment, produce the well-known concentric 
rings. The spherocrystals of inulin, in hke manner, 
shew, like the “calcospherites” of Harting (Fig. 
208), a concentric structure which in all likelihood 
has had no causative impulse save from within. 

The striation, or concentric lamellation, of the scales and 
otoliths of fishes has been much 
employed of recent years as a 
trustworthy and unmistakeable 
mark of the fish’s age. There 
are difficulties in the way of 
accepting this hypothesis, not the 
least of which is the fact that 
the otolith-zones, for instance, 
Fig. 209. Otoliths of Plaice, showing are extremely well marked even 


four zones or “age-rings.”” (After jn the case of some fishes which 
Wallace.) 


Fig. 208. 


spend their lives in deep water, 


* Cf. Biitschli, Ueber die Herstellung kiinstlicher Stirkek6rner oder von 
Spharokrystallen der Stiirke, Verh. nat. med. Ver. Heidelberg, v, pp. 457-472, 1896. 
+ Untersuchungen iiber die Stdrkekérner, Jena, 1905. 


1x] OF FISHES’ SCALES AND OTOLITHS 433 


where the temperature and other physical conditions shew little 
or no appreciable fluctuation with the seasons of the year. 
There are, on the other hand, phenomena which seem strongly 
confirmatory of the hypothesis: for instance the fact (if it 
be fully established) that in such a fish as the cod, zones of 
growth, identical in number, are found both on the scales and 
in the otoliths*. The subject has become a much debated one, 
and this is not the place for its discussion; but it is at least 
obvious, with the Liesegang phenomenon in view, that we have 
no right to asswme that an appearance of rhythm and periodicity 
in structure and growth is necessarily bound up with, and 
indubitably brought about by, a periodic recurrence of particular 
external conditions, 

But while in the Liesegang phenomenon we have rhythmic 
precipitation which depends only on forces intrinsic to the system, 
and is independent of any corresponding rhythmic changes in 
temperature or other external conditions, we have not far to seek 
for instances of chemico-physical phenomena where rhythmic 
alternations of appearance or structure are produced in close 
relation to periodic fluctuations of temperature. A well-known 
instance is that of the Stassfurt deposits, where the rock-salt 
alternates regularly with thin layers of “anhydrite,” or (in 
another series of beds) with “polyhalite+’: and where these 
zones are commonly regarded as marking years, and their 
alternate bands as having been formed in connection with the 
seasons. A discussion, however, of this remarkable and significant 
phenomenon, and of how the chemist explains it, by help of the 
“phase-rule,” in connection with temperature conditions, would 
lead us far beyond our scope f. 


We now see that the methods by which we attempt to study 
the chemical or chemico-physical phenomena which accompany 
the development of an inorganic concretion or spicule within the 


* Cf. Winge, Meddel. fra Komm. for Havunderségelse (Fiskeri), tv, p. 20, Copen- 
hagen, 1915. 

.+ The anhydrite is sulphate of lime (CaSO,); the polyhalite is a triple sulphate 
of lime, magnesia and potash (2CaSO,. MgSO,. K,SO,+ 2H,0). 

t Cf. van’t Hoff, Physical Chemistry in the Service of the Sciences, p. 99 seq, 
Chicago, 1903. 


T. G. 28 


434 ON CONCRETIONS, SPICULES, ETC. [CH. 


body of an organism soon introduce us to a multitude of kindred 
phenomena, of which our knowledge is still scanty, and which we 
must not attempt to discuss at greater length. As regards our 
main point, namely the formation of spicules and other elementary 
skeletal forms, we have seen that certain of them may be safely 
ascribed to simple precipitation or crystallisation of inorganic 
materials, in ways more or less modified by the presence of 
albuminous or other colloid substances. The effect of these 
latter is found to be much greater in the case of some crystallisable - 
bodies than in others. For instance, Harting, and Rainey also, 
found as a rule that calcium oxalate was much less affected by 
a colloid medium than was calcium carbonate; it shewed in 
their hands no tendency to form rounded concretions or “calco- 
spherites” in presence of a colloid, but continued to crystallise, 
either normally, or with a tendency to form needles or raphides. 
It is doubtless for this reason that, as we have seen, crystals of 
calcium oxalate are so common in the tissues of plants, while 
those of other calcium salts are rare. But true calcospherites, 
or spherocrystals, of the oxalate are occasionally found, for 
instance in certain Cacti, and Biitschli* has succeeded in making 
them artificially in Harting’s usual way, that is to say by crystal- 
lisation in a colloid medium. 

There link on to these latter observations, and to the statement 
already quoted that calcareous deposits are associated with the 
dead products rather than with the living cells of the organism, 
certain very interesting facts in regard to the solubility of salts 
in colloid media, which have been made known to us of late, and 
which go far to account for the presence (apart from the form) 
of calcareous precipitates within the organismy. It has been 
shewn, in the first place, that the presence of albumin has a notable 
effect on the solubility in a watery solution of calcium salts, 
increasing the solubility of the phosphate in a marked degree, 
and that of the carbonate in still greater proportion; but the 

* Spharocrystalle von Kalkoxalat bei Kakteen, Ber. d. d. Bot. Gesellsch. 
p. 178, 1885. 

+ Pauli, W. u. Samec, M., Ueber Loslichkeitsbeeinfliissung von Elektrolyten 
durch Eiweisskérper, Biochem. Zeitschr. xvu, p. 235, 1910. Some of these results 


were known much earlier; cf. Fokker in Pfliiger’s Archiv, vu, p. 274, 1873; also 
Irvine and Sims Woodhead, op. cit. p. 347. 


IX] OF SOLUBILITY IN COLLOID MEDIA 435 


sulphate is only very little more soluble in presence of albumin 
than in pure water, and the rarity of its occurrence within the 
organism is so far accounted for. On the other hand, the bodies 
derived from the breaking down of the albumins, their “catabolic” 
products, such as the peptones, etc., dissolve the calcium salts to 
a much less degree than albumin itself; and in the case of the 
phosphate, its solubility in them is scarcely greater than in water. 
The probability is, therefore, that the actual precipitation of the 
calcium salts is not due to the direct action of carbonic acid, etc. 
on a more soluble salt (as was at one time believed); but to cata- 
bole changes in the proteids of the organism, which tend to throw 
down the salts already formed, which had remained hitherto in 
albuminous solution. The very slight solubility of calcium phos- 
phate under such circumstances accounts for its predominance 
in, for instance, mammalian bone*; and wherever, in short, the 
supply of this salt has been available to the organism. 

To sum up, we see that, whether from food or from sea-water, 
calcium sulphate will tend to pass but little into solution in the 
albuminoid substances of the body: calcium carbonate will enter 
more freely, but a considerable part of it will tend to remain in 
solution: while calcium phosphate will pass into solution in 
considerable amount, but will be almost whcelly precipitated 
again, as the albumin becomes broken down in the normal process 
of metabolism. 

We have still to wait for a similar and equally illuminating 
study of the solution and precipitation of szlica, in presence of 
organic colloids. 


From the comparatively small group of inorganic formations 
which, arising within living organisms, owe their form solely to 
precipitation or to crystallisation, that is to say to chemical or other 
molecular forces, we shall presently pass to that other and larger 
group which appear to be conformed in direct relation to the forms 
and the arrangement of the cells or other protoplasmic elements 7. 


* Which, in 1000 parts of ash, contains about 840 parts of phosphate and 
76 parts of calcium carbonate. 
+ Cf. Dreyer, Fr., Die Principien der Geriistbildung bei Rhizopoden, Spongien 
und Echinodermen, Jen. Zeitschr. xxvi, pp. 204-468, 1892. 
28—2 


436° ON CONCRETIONS, SPICULES, . (CH. 


The two principles of conformation are both illustrated in the 
spicular skeletons of the Sponges. 

In a considerable number, but withal a minority of cases, the 
form of the sponge-spicule may be deemed sufficiently explained 
on the lines of Harting’s and Rainey’s experiments, that 1s to say 
as the direct result of chemical or physical phenomena associated 
with the deposition of lime or of silica in presence of colloids*. 
This is the case, for instance, with various small spicules of a 
globular or spheroidal form, formed of amorphous silica, con- 


Fig. 210. Close-packed calcospherites, or so-called “spicules,” 
of Astrosclera. (After Lister.) 


centrically striated within, and often developing irregular knobs 
or tiny tubercles over their surfaces. In the aberrant sponge 
Astrosclerat, we have, to begin with, rounded, striated discs or 
globules, which in like manner are nothing more or less than the 


* In an anomalous and very remarkable Australian sponge, just described by 
Professor Dendy (Nature, May 18, 1916, p. 253) under the name of Collosclerophora, 
the spicules are “gelatinous,” consisting of a gel of colloid silica with a high 
percentage of water. It is not stated whether an organic colloid is present together 
with the silica. These gelatinous spicules arise as exudations on the outer surface 
of cells, and come to lie in intercellular spaces or vesicles. 

7 Lister, in Willey’s Zoological Results, pt 1v. p. 459, 1900. 


Ix] AND SPICULAR SKELETONS 437 


“calcospherites” of Harting’s experiments; and as these grow 
they become closely aggregated together (Fig. 210), and assume an 
angular, polyhedral form, once more in complete accordance with 
the results of experiment*. Again, in many Monaxonid sponges, 
we have irregularly shaped, or branched spicules, roughened or 
tuberculated by secondary superficial deposits, and reminding one 
of the spicules of some Alcyonaria. These also must be looked 
upon as the simple result of chemical deposition, the form of the 
deposit being somewhat modified in conformity with the surround- 
ing tissues, just as in the simple experiment the form of the con- 
cretionary precipitate is affected by the heterogeneity, visible or 
invisible, of the matrix. Lastly, the simple needles of amorphous 
silica, which constitute one of the commonest types of spicule, 
call for little in the way of explanation; they are accretions or 
deposits about a linear axis, or fine thread of organic material, 
just as the ordinary rounded calcospherite is deposited about 
some minute point or centre of crystallisation, and as ordinary 
crystalhsation is often started by a particle of atmospheric dust; 
in some cases they also, like the others, are apt to be roughened 
by more irregular secondary deposits, which probably, as in 
Harting’s experiments, appear im this irregular form when the 
supply of material has become relatively scanty. 


Our few foregoing examples, diverse as they are in look and 
kind and ranging from the spicules of Astrosclera or Aleyonium 
to the otoliths of a fish, seem all to have their free origin in some 
larger or smaller fluid-containing space, or cavity of the body: 
pretty much as Harting’s calcospheres made their appearance in 
the albuminous content of a dish. But we now come at last to 
a much larger class of spicular and skeletal structures, for whose 
regular and often complex forms some other explanation than the 
intrinsic forces of crystallisation or molecular adhesion is mani- 
festly necessary. As we enter on this subject, which is certainly 
no small or easy one, it may conduce to simplicity, and to brevity, 


* The peculiar spicules of Astrosclera are now said to consist of spherules, or 
calcospherites, of aragonite, spores of a certain red seaweed forming the nuclei, 
or starting-points, of the concretions (R. Kirkpatrick, Proc. R. S. Lxxxtv (B), 
p. 579, 1911. 


438 ON CONCRETIONS, SPICULES, [CH. 


if we try to make a rough classification, by way of forecast, of 
the chief conditions which we are likely to meet with. 

Just as we look upon animals as constituted, some of a vast 
number of cells, and others of a single cell or of a very few, and 
just as the shape of the former has no longer a visible relation to 
the individual shapes of its constituent cells, while in the latter 
it is cell-form which dominates or is actually equivalent to the 
form of the organism, so shall we find it to be, with more or less 
exact analogy, in the case of the skeleton. For example, our own 
skeleton consists of bones, in the formation of each of which a 
vast number of minute living cellular elements are necessarily 
concerned; but the form and even the arrangement of these 
bone-forming cells or corpuscles are monotonously simple, and we 
cannot find in these a physical explanation of the outward and 
visible configuration of the bone. It is as part of a far larger 
field of force,—in which we must consider gravity, the action. of 
various muscles, the compressions, tensions and bending moments 
due to variously distributed loads, the whole interaction of a very 
complex mechanical system,—that we must explain (if we are to 
explain at all) the configuration of a bone. 

Tn contrast to these massi¥e skeletons, or constituents of a 
skeleton, we have other skeletal elements whose whole magnitude, 
or whose magnitude in some dimension or another, is commensurate 
with the magnitude of a single living cell, or (as comes to very 
much the same thing) is comparable to the range of action of the 
molecular forces. Such is the case with the ordinary spicules of 
a sponge, with the delicate skeleton of a Radiolarian, or with the 
denser and robuster shells of the Foraminifera. The effect of 
scale, then, of which we had so much to say in our introductory 
chapter on Magnitude, is bound to be apparent in the study of 
skeletal fabrics, and to lead to essential differences between the 
big and the little, the massive and the minute, in regard to their 
controlling forces and their resultant forms. And if all this be 
so, and if the range of action of the molecular forces be in truth 
the important and fundamental thing, then we may somewhat 
extend our statement of the case, and include in it not only 
association with the living cellular elements of the body, but also 
association with any bubbles, drops, vacuoles or vesicles which 


Ix] AND SPICULAR SKELETONS 439 


may be comprised within the bounds of the organism, and which 
are (as their names and characters connote) of the order of 
magnitude of which we are speaking. 

Proceeding a little farther in our classification, we may conceive 
each little skeletal element to be associated, in one case, with 
a single cell or vesicle, and in another with a cluster or “system” 
of consociated cells. In either case there are various possibilities. 
For instance, the calcified or other skeletal material may tend 
to overspread the entire outer surface of the cell or cluster of cells, 
and so tend accordingly to assume some configuration comparable 
to that of a fluid drop or of an aggregation of drops; this, in brief, 
is the gist and essence of our story of the foraminiferal shell. 
Another common, but very different condition will arise if, in the 
case of the cell-aggregates, the skeletal material tends to accumulate 
in the interstices between the cells, in the partition-walls which 
separate them, or in the still more restricted distribution indicated 

_by the lines of junction between these partition-walls. Conditions 

such as these will go a very long way to help us in our under- 
standing of many sponge-spicules and of an immense variety of 
radiolarian skeletons. And lastly (for the present), there is a 
possible and very interesting case of a skeletal element associated 
with the surface of a cell, not so as to cover it like a shell, but 
only so as to pursue a course of its own within it, and subject to 
the restraints imposed by such confinement to a curved and 
limited surface. With this curious condition we shall deal 
immediately. 

This preliminary and much simplified classification of skeletal 
forms (as is evident enough) does not pretend to completeness. 
It leaves out of account some kinds of conformation and con- 
figuration with which we shall attempt to deal, and others which 
we must perforce omit. But nevertheless it may help to clear 
or to mark our way towards the subjects which this chapter has 
to consider, and the conditions by which they are at least partially 
defined. . 


Among the several possible, or conceivable, types of microscopic 
skeletons let us choose, to begin with, the case of a spicule, more 
or less simply linear as far as its intrinsic powers of growth are 


440 ON CONCRETIONS, SPICULES, ETC. [CH. 


concerned, but which owes its now somewhat complicated form 
to a restraint imposed by the individual cell to which it is confined, 
and within whose bounds it is generated. The conception of a 
spicule developed under such conditions we owe to a distinguished 
physicist, the late Professor G. F. FitzGerald. 

Many years ago, Sollas pointed out that if a spicule begin to 
grow In some particular way, presumably under the control or 
constraint imposed by the organism, it continues to grow by 
further chemical deposition in the same form or direction even 
after it has got beyond the boundaries of the organism or its 
cells. This phenomenon is what we see in, and this imperfect 
explanation goes so far to account for, the continued growth in 
straight lines of the long calcareous spines of Globigerina or 
Hastigerina, or the similarly radiating but siliceous spicules of 
many Radiolaria. In physical language, if our crystalline 
structure has once begun to be laid down in a definite orientation, 
further additions tend to accrue in a like regular fashion and in 
an identical direction; and this corresponds to the phenomenon 
of so-called “orientirte Adsorption,” as described by Lehmann. 

In Globigerina or in Acanthocystis the long needles grow out 
freely into the surrounding medium, with nothing to impede their 
rectilinear growth and their approximately radiate distribution. 
But let us consider some simple cases to illustrate the forms which 
a spicule will tend to assume when, striving (as it were) to grow 
straight, it comes under the influence of some simple and constant 
restraint or compulsion. 

If we take any two points on some curved surface, such as 
that of a sphere or an ellipsoid, and imagine a string stretched 
between them, we obtain what is known in mathematics as a 
“geodetic”? curve. It is the shortest ine which can be traced 
between the two points, upon the surface itself; and the most 
familiar of all cases, from which the name is derived, is that curve 
upon the earth’s surface which the navigator learns to follow in 
the practice of “great-circle sailing.” Where the surface is 
spherical, the geodetic is always literally a “great circle,” a circle, 
that is to say, whose centre is the centre of the sphere. If instead 
of a sphere we be dealing with an ellipsoid, the geodetic becomes 
a variable figure, according to the position of our two points. 


Ix] OF INTRACELLULAR SPICULES 44] 


For obviously, 1f they le in a line perpendicular to the long axis 
of the ellipsoid, the geodetic which connects them is a circle, also 
perpendicular to that axis; and if they le in a line parallel to 
the axis, their geodetic is a portion of that ellipse about which 
the whole figure is a solid of revolution. But if our two points 
he, relatively to one another, in any other direction, then their 
geodetic is part of a spiral curve in space, winding over the surface 
of the ellipsoid. 

To say, as we have done, that the geodetic is the shortest line 
between two points upon the surface, is as much as to say that 
it is a projection of some particular straight line upon the surface 
in question; and it follows that, if any linear body be confined 
to that surface, while retaining a tendency to grow by successive 
increments always (save only for its confinement to that surface) 
in a straight line, the resultant form which it will assume will be 
that of a geodetic. In mathematical language, it is a property 
of a geodetic that the plane of any two consecutive elements is 
a plane perpendicular to that in which the geodetic lies; or, in 
simpler words, any two consecutive elements le in a straight line 
in the plane of the surface, and only diverge from a straight line 
in space by the actual curvature of the surface to which they are 
restrained. 

Let us now imagine a spicule, whose natural tendency is to 
grow into a straight linear element, either by reason of its own 
molecular anisotropy, or because it is deposited about a thread- 
like axis; and let us suppose that it is confined either within a 
cell-wall or in adhesion thereto: it at once follows that its lne 
of growth will be simply a geodetic to the surface of the cell. 
And if the cell be an imperfect sphere, or a more or less regular 
ellipsoid, the spicule will tend to grow into one or other of three 
forms: either a plane curve of circular arc; or, more commonly, 
a plane curve which is a portion of an ellipse; or, most commonly 
of all, a curve which is a portion of a spiral in space. In the 
latter case, the number of turns of the spiral will depend, not only 
on the length of the spicule, but on the relative dimensions of 
the ellipsoidal cell, as well as upon the angle by which the spicule 
is inclined to the ellipsoid axes; but a very common case will 
probably be that in which the spicule looks at first sight to be 


442 ON CONCRETIONS, SPICULES, ETC. [CH. 


a plane C-shaped figure, but is discovered, on more careful inspec- 
tion, to lie not in one plane but in a more complicated spiral twist. 


IC 


Fig. 211. 


LO YG 


Sponge and Holothurian spicules. 


This investigation includes a series of forms which are abundantly 


represented among 


actual sponge-spicules, as illustrated in 


Figs. 211 and 212. If the spicule be not restricted 


VO. ito, dimear growth, but have a tendency to ex- 
pand, or to branch out from a main axis, we shall 
obtain a series of more complex figures, all related 

_/ to the geodetic system of curves. A very simple 


Fig. 212. 


case will arise where the spicule occupies, in the 
first instance, the axis of the containing cell, 


and then, on reaching its boundary, tends to branch or 
spread outwards. We shall now get various figures, in some 


Fig. 213. An “‘amphidisc”’ 
of Hyalonema. 


of which the spicule will appear as an axis 
expanding into a disc or wheel at either 
end; and in other cases, the terminal dise 
will be replaced, or represented, by a series 
of rays or spokes, with a reflex curvature, 
corresponding to the spherical or ellipsoid 
curvature of the surface of the cell. Such 
spicules as these are again exceedingly 
common among various sponges (Fig. 213). 

Furthermore, if these mechanical methods 
of conformation, and others like to these, 
be the true cause of the shapes which the 
spicules assume, it is plain that the pro- 


duction of these spicular shapes is not a specific function of 
sponges or of any particular sponge, but that we should expect 


IX] OF INTRACELLULAR SPICULES 443 


the same or very similar phenomena to occur in other organisms, 
wherever the conditions of inorganic secretion within closed cells 
was very much the same. As a matter of fact, in the group of 
Holothuroidea, where the formation of intracellular spicules is a 
characteristic feature of the group, all the principal types of 
conformation which we have just described can be closely 
paralleled. Indeed in many cases, the forms of the Holothurian 
spicules are identical and indistinguishable from those of the 
sponges*. But the Holothurian spicules are composed of calcium 
carbonate while those which we have just described in the case 
of sponges are usually, if not always, siliceous: this being just 
another proof of the fact that in such cases the form of the 
spicule is not due to its chemical nature or molecular structure, 
but to the external forces to which, during its growth, the 
spicule is submitted. 


So much for that comparatively limited class of sponge- 
_ spicules whose forms seem capable of explanation on the hypothesis 
that they are developed within, or under the restraint imposed by, 
the surface of a cell or vesicle. Such spicules are usually of small 
size, as well as of comparatively simple form; and they are greatly 
outstripped in number, in size, and in supposed importance as 
guides to zoological classification, by another class of spicules. 
This new class includes such as we have supposed to be capable 
of explanation on the assumption that they develop in association 
(of some sort or another) with the lines of junction of contiguous 
cells. They include the triradiate spicules of the calcareous 
sponges, the quadriradiate or “ tetractinellid”’ spicules which occur 
in the same group, but more characteristically in certain siliceous 
sponges known as the Tetractinellidae, and lastly perhaps (though 
these last are admittedly somewhat harder to understand) the 
six-rayed spicules of the Hexactinellids. 

The spicules of the calcareous sponges are commonly tri- 
radiate, and the three radii are usually inclined to one another 
at equal, or nearly equal angles; in certain cases, two of the 
three rays are nearly in a straight line, and at right angles to the 


* See for instance the plates in Théel’s Monograph of the Challenger Holo- 
thuroidea; also Sollas’s Tetractinellida, p. 1xi. 


444 ON CONCRETIONS, SPICULES, ETC. [CH. 


third*. They are seldom in a plane, but are usually inclined to 
one another in a solid, trihedral angle, not easy of precise measure- 
ment under the microscope. The three rays are very often 
supplemented by a fourth, which is set tetrahedrally, making, that 
is to say, coequal angles with the other three. The calcareous 
spicule consists mainly of carbonate of lime, in the form of calcite, 
with (according to von Ebner) some admixture of soda and* 
magnesia, of sulphates and of water. According to the same 
writer (but the fact, though it would seem easy to test, is still 
disputed) there is no organic matter in the spicule, either in the 
form of an axial filament or otherwise, and the appearance of 
stratification, often simulating the presence of an axial fibre, is 
due to “mixed crystallisation” of the various constituents. The 
spicule is a true crystal, and therefore its existence and its form 
are primardy due to the molecular forces of crystallisation ; more- 
over it is a single crystal and not a group of crystals, as is at once 
seen by its behaviour in polarised light. But its axes are not 
crystalline axes, and its form neither agrees with, nor in any way 
resembles, any one of the many polymorphic forms in which 
calcite is capable of crystallising. It is as though it were carved 
out of a solid crystal; it is, in fact, a crystal under restraint, 
a crystal growing, as it were, in an artificial mould; and this 
mould is constituted by the surrounding cells, or structural 
vesicles of the sponge. : 

We have already studied in an elementary way, but amply 
for our present purpose, the manner in which three or more cells, 
or bubbles, tend to meet together under the influence of surface- 
tension, and also the outwardly similar phenomena which may be 
brought about by a uniform distribution of mechanical pressure. 
We have seen that when we confine ourselves to a plane assemblage 
of such bodies, we find them meeting one another in threes; that 
in a section or plane projection of such an assemblage we see the. 
partition-walls meeting one another at equal angles of 120°; that 
when the bodies are uniform in size, the partitions are straight 
lines, which combine to form regular hexagons; and that when 


* For very numerous illustrations of the triradiate and quadriradiate spicules 
of the calcareous sponges, see (int. al.), papers by Dendy (Q. J. M. S. xxxv, 1893), 
Minchin (P. Z. S. 1904), Jenkin (P. Z. 8. 1908), ete. 


Ix] OF THE SKELETON OF SPONGES 445 


the bodies are unequal in size, the partitions are curved, and 
combine to form other and less regular polygons. It is plain, 
accordingly, that in any flattened or stratified assemblage of such 
cells, a solidified skeletal deposit which originates or accumulates 
either between the cells or within the thickness of their mutual 
partitions, will tend to take the form of triradiate bodies, whose - 
rays (in a typical case) will be set at equal angles of 120°(Fig. 214, F). 
And this latter condition of equality will be open to modification 


Fig. 214. Spicules of Grantia and other calcareous sponges, 
(After Haeckel.) 


in various ways. It will be modified by any inequality in the 
specific tensions of adjacent cells; as a special case, it will be apt 
to be greatly modified at the surface of the system, where a spicule 
happens to be formed in a plane perpendicular to the cell-layer, 
so that one of its three rays lies between two adjacent cells and 
the other two are associated with the surface of contact between 
the cells and the surrounding medium; in such a case (as in the 
cases considered in connection with the forms of the cells themselves 


446 ON CONCRETIONS, SPICULES, ETC. [CH. 


on p. 314), we shall tend to obtain a spicule with two equal angles 
and one unequal (Fig. 214, A, C). In the last case, the two outer, 
or superficial rays, will tend to be markedly curved. Again, the 
equiangular condition will be departed from, and more or less 
curvature will be imparted to the rays, wherever the cells of the 
system cease to be uniform in size, and when the hexagonal 
symmetry of the system is lost accordingly. Lastly, although we 
speak of the rays as meeting at certain definite angles, this state- 
ment applies to their axes, rather than to the rays themselves. 
For, if the triradiate spicule be developed in the interspace between 
three juxtaposed cells, it is obvious that its sides will tend to be 
concave, for the interspace between our three contiguous equal 
circles is an equilateral, curvilinear triangle; and even if our 
spicule be deposited, not in the space between our three cells, 
but in the thickness of the intervening wall, thenewe may recollect 
(from p. 297) that the several partitions never actually meet at 
sharp angles, but the angle of contact is always bridged over by 
a small accumulation of material (varying in amount according 
to its fluidity) whose boundary takes the form of a circular arc, 
and which constitutes the “bourrelet” of Plateau. 

In any sample of the triradiate spicules of Grantia, or in any 
series of careful drawings, such as those of Haeckel among others, 
we shall find that all these various configurations are precisely 
and completely illustrated. 

The tetrahedral, or rather tetractinellid, spicule needs no 
explanation in detail (Fig. 214, D, Z). For just as a triradiate 
spicule corresponds to the case of three cells in mutual contact, 
so does the four-rayed spicule to that of a solid aggregate of four 
cells: these latter tending to meet one another in a tetrahedral 
system, shewing four edges, at each of which four surfaces meet, 
the edges being inclined to one another at equal angles of about 
109°. And even in the case of a single layer, or superficial layer, 
of cells, if the skeleton originate in connection with all the edges 
of mutual contact, we shall, in complete and typical cases, have 
a four-rayed spicule, of which one straight limb will correspond 
to the line of junction between the three cells, and the other three 
hmbs (which will then be curved limbs) will correspond to the edges 
where two cells meet one another on the surface of the system. 


1X] OF THE SKELETON OF SPONGES 447 


But if such a physical explanation of the forms of our spicules 
is to be accepted, we must seek at once for some physical agency 
by which we may explain the presence of the solid material just 
at the junctions or interfaces of the cells, and for the forces by 
which it is confined to, and moulded to the form of, these inter- 
cellular or interfacial contacts. It is to Dreyer that we chiefly 
owe the physical or mechanical theory of spicular conformation 
which I have just described,—a theory which ultimately rests 
on the form assumed, under surface-tension, by an aggregation 
of cells or vesicles. But this fundamental point being granted, 
we have still several possible alternatives by which to explain the 
details of the phenomenon. 

Dreyer, if I understand him aright, was content to assume that 
the solid material, secreted or excreted by the organism, accumu- 
lated in the interstices between the cells, and was there subjected 
to mechanical pressure or constraint as the cells got more and 
more crowded together by their own growth and that of the 
system generally. As far as the general form of the spicules goes, 
such explanation is not inadequate, though under it we may have 
to renounce some of our assumptions as to what takes place at 
the outer surface of the system. 

But‘in all (or most) cases where, but a few years ago, the 
concepts of secretion or excretion seemed precise enough, we are 
now-a-days inclined to turn to the phenomenon of adsorption as 
a further stage towards the elucidation of our facts. Here we 
have a case in point. In the tissues of our sponge, wherever two 
cells meet, there we have a definite surface of contact, and there 
accordingly we have a manifestation of surface-energy; and the 
concentration of surface-energy will tend to be a maximum at 
the lanes or edges whereby the three, or four, such surfaces are 
conjoined. Of the micro-chemistry of the sponge-cells our 
ignorance is great; but (without venturing on any hypothesis 
involving the chemical details of the process) we may safely assert 
that there is an inherent probability that certain substances will 
tend to be concentrated and ultimately deposited just in these lines 
of intercellular contact and conjunction. In other words, adsorp- 
tive concentration, under osmotic pressure, at and in the surface- 
film which constitutes the mutual boundary between contiguous 


448 ON CONCRETIONS, SPICULES, ETC. [cH. 


cells, emerges as an alternative (and, as it seems to me, a highly 
preferable alternative) to Dreyer’s conception of an accumulation 
under mechanical pressure in the vacant spaces left between one 
cell and another. 

But a purely chemical, or purely molecular adsorption, is not 
the only form of the hypothesis on which we may rely. For 
from the purely physical point of view, angles and edges of contact 
between adjacent cells will be loci in the field of distribution of 
surface-energy, and any material particles whatsoever will tend 
to undergo a diminution of freedom on entering one of those 
boundary regions. In a very simple case, let us imagine a couple 
of soap bubbles in contact with one another. Over the surface 
of each bubble there glide in every direction, as usual, a multitude 
of tiny bubbles and droplets; but as soon as these find their way 
into the groove or re-entrant angle between the two bubbles, 
there their freedom of movement is so far restrained, and out of 
that groove they have little or no tendency to emerge. A cognate 
phenomenon is to be witnessed in microscopic sections of steel or 
other metals. Here, amid the “crystalline” structure of the 
metal (where in cooling its imperfectly homogeneous material has 
developed a cellular structure, shewing (in section) hexagonal or 
polygonal contours), we can easily observe, as Professot Peddie 
has shewn me, that the little particles of graphite and other 
foreign bodies common in the matrix, have tended to aggregate 
themselves in the walls and at the angles of the polygonal 
cells—this being a direct result of the diminished freedom 
which the particles undergo on entering one of these boundary 
regions*. 

It is by a combination of these two principles, chemical adsorp- 
tion on the one hand, and physical quasi-adsorption or concentration 
of grosser particles on the other, that I conceive the substance 
of the sponge-spicule to be concentrated and aggregated at the 
cell boundaries; and the forms of the triradiate and tetractinellid 
spicules are in precise conformity with this hypothesis. A few 
general matters, and a few particular cases, remain to be con- 
sidered. 

It matters little or not at all, for the phenomenon in question, 


* Cf. again Bénard’s Tourbillons cellulaires, Ann. de Chimie, 1901, p. 84. 


Ix] _ OF THE SKELETON OF SPONGES 449 


what is the histological nature or “ grade” of the vesicular structures 
on which it depends. In some cases (apart from sponges), they 
may be no more than the little alveoli of the intracellular proto- 
plasmic network, and this would seem to be the case at least in 
one known case, that of the protozoan Entosolenia aspera, in which, 
within the vesicular protoplasm of the single cell, Mobius has 
described tiny spicules in the shape of little tetrahedra with 
concave sides. It 1s probably also the case in the small beginnings 
of the Echinoderm spicules, which are likewise intracellular, and 
are of similar shape. In the case of our sponges we have many 
varying conditions, which we need not attempt to examine in 
detail. In some cases there is evidence for believing that the 
spicule is formed at the boundaries of true cells or histological 
units. But in the case of the larger triradiate or tetractinellid 
spicules of the sponge-body, they far surpass in size the actual 
“cells”; we find them lying, regularly and symmetrically 
arranged, between the “pore-canals” or “ciliated chambers,” 
and it is in conformity with the shape and arrangement of these 
rounded or spheroidal structures that their shape is assumed. 

Again, it is not necessarily at variance with our hypothesis 
to find that, in the adult sponge, the larger spicules may greatly 
outgrow the bounds not only of actual cells but also of the 
ciliated chambers, and may even appear to project freely from the 
surface of the sponge. For we have already seen that the spicule 
is capable of growing, without marked change of form, by further 
deposition, or crystallisation, of layer upon layer of calcareous 
molecules, even in an artificial solution; and we are entitled to 
believe that the same process may be carried on in the tissues of 
the sponge, without greatly altering the symmetry of the spicule, 
long after it has established its characteristic form of a system of 
slender trihedral or tetrahedral rays. 

Neither is it of great importance to our hypothesis whether 
the rayed spicule necessarily arises as a single structure, or does 
so from separate minute centres of aggregation. Minchin has 
shewn that, in some cases at least, the latter is the case; the 
spicule begins, he tells us, as three tiny rods, separate from one 
another, each developed in the interspace between two sister- 
cells, which are themselves the results of the division of one of a 


T. G. 29 


450 ON CONCRETIONS, SPICULES, ETC. [CH. 


little trio of cells; and the little rods meet and fuse together while 
still very minute, when the whole spicule is only about 1, of a 
millimetre long. At this stage, it is interesting to learn that the 
spicule is non-crystalline; but the new accretions of calcareous 
matter are soon deposited in crystalline form. 

This observation threw considerable difficulties in the way of 
former mechanical theories of the conformation of the spicule, and 
was quite at variance with Dreyer’s theory, according to which 
the spicule was bound to begin from a central nucleus coinciding 
with the meeting-place of the three contiguous cells, or rather the 
interspace between them. But the difficulty is removed when we 
import the concept of adsorption; for by this agency it is natural 
enough, or conceivable enough, that the process of deposition 
should go on at separate parts of a common system of surfaces; 
and if the cells tend to meet one another by their interfaces before 
these interfaces extend to the angies and so complete the polygonal | 
cell, it is again conceivable and natural that the spicule should 
first arise in the form of separate and detached limbs or rays. 

Among the tetractinellid sponges, 


aN ; whose spicules are composed of amor- 
6h T phous silica or opal, all or most of the 
Hoa 6 } © above-described main types of spicule 


occur, and, as the name of the group 
implies, the four-rayed, tetrahedral 
spicules are especially represented. A 
somewhat frequent type of spicule is 
one in which one of the four rays is 
greatly developed, and the other three 
constitute small prongs diverging at 
equal angles from the main or axial 
ray. In all probability, as Dreyer 
suggests, we have here had to do with 
a group of four vesicles, of which 
three were large and co-equal, while a 
fourth and very much smaller one lay 
Fig. 215. . Spicules of tetracti- ghoyve and between the other three. 

nellid sponges (after Sollas). ieee hee we haveshiae 

n certain cases w 


a-e, anatriaenes; d-f, pro- , 
triaenes. wise one large and three much smaller 


Ix] OF THE SKELETON OF SPONGES 451 


rays, the latter are recurved, as in Fig. 215. This type, save for 
the constancy of the number of rays, and the limitation of the 
terminal ones to three, and save also for the more important 
difference that they occur only at one and not at both ends of 
the long axis, is similar to the type of spicule illustrated in 
Fig. 213, which we have explained as being probably developed 
within an oval cell, by whose walls its branches have been con- 
formed to geodetic curves. But it is much more probable that 
we have here to do with a spicule developed in the midst of a 
group of three coequal and more or less elongated or cylindrical 
cells or vesicles, the long axial ray corresponding to their common 


aX 


? 


Fig. 216. Various holothurian spicules. (After Theel.) 


line of contact, and the three short rays having each lain in the 
surface furrow between two out of the three adjacent cells. 

Just as in the case of the little curved or S-shaped spicules, 
formed apparently within the bounds of a single cell, so also in 
the case of the larger tetractinellid and analogous types do we 
find among the Holothuroidea the same configurations reproduced 
as we have dealt with in the sponges. The holothurian spicules 
are a little less neatly formed, a little rougher, than the sponge- 
spicules; and certain forms occur among the former group which 
do not present themselves among the latter; but for the most 
part a community of type is obvious and striking (Fig. 216). 

A curious and, physically speaking, strictly analogous forma- 
tion to the tetrahedral spicules of the sponges is found in the 

292 


452 ON CONCRETIONS, SPICULES, ETC. - CH: 


spores of a certain little group of parasitic protozoa, the Actino- 
myxidia. These spores are formed from clusters of six cells, 
of which three come to constitute the capsule of the spore; and 
this capsule, always triradiate in its symmetry, is in some species 
drawn out into long rays, of which one constitutes a straight 
central axis, while the others, coming off from it at equal angles, 
are recurved in wide circular arcs. The account given of the 
development of this structure by its discoverers* is somewhat 
obscure to me, but I think that, on physical grounds, there can 
be no doubt whatever that the quadriradiate capsule has been 
somehow modelled upon a group of three surrounding cells, its 
axis lying between the three, and its three radial arcs occupying 
the furrows between adjacent pairs. 


4 


Fig. 217. Spicules of hexactinellid sponges. (After F. E. Schultze.) 


The typically six-rayed siliceous spicules of the hexactinellid 
sponges, while they are perhaps the most regular and beautifully 
formed spicules to be found within the entire group, have been 
found very difficult to explain, and Dreyer has confessed his 
complete inability to account for their conformation. But, 
though it is doubtless only throwing the difficulty a little further 
back, we may so far account for them by considering that the 
cells or vesicles by which they are conformed are not arranged in 


* Léger, Stole and others, in Dofleim’s Lehrbuch d. Protozoenkunde, 1911, 
p. 912. 


Ix] OF THE SKELETON OF SPONGES 453 


what is known as “closest packing,” but in linear series; so that in 
their arrangement, and by their mutual compression, we tend to 
get a pattern, not of hexagons, but of squares: or, looking to 
the solid, not of dodecahedra but of cubes or parallelopipeda. 
This indeed appears to be the case, not with the individual cells 
(in the histological sense), but with the larger units or vesicles 
which make up the body of the hexactinellid. And this being 
so, the spicules formed between the linear, or cubical series of 
vesicles, will have the same tendency towards a “hexactinellid” 
shape, corresponding to the angles and adjacent edges of a system 
of cubes, as in our former case they had to a triradiate or a 
tetractinellid form, when developed in connection with the angles 
and edges of a system of hexagons, or a system of dodecahedra. 

Histologically, the case is illustrated by a well-known pheno- 
menon in embryology. In the segmenting ovum, there is a 
tendency for the cells to be budded off in linear series; and so 
_ they often remain, in rows side by side, at least for a considerable 
time and during the course of several consecutive cell divisions. 
Such an arrangement constitutes what the embryologists call the 
“radial type” of segmentation*. But in what is described as the 
“spiral type” of segmentation, it is stated that, as soon as the 
first horizontal furrow has divided the cells into an upper and 
a lower layer, those of “the upper layer are shifted in respect 
to the lower layer, by means of a rotation about the vertical 
axist.” It is, of course, evident that the whole process is 
merely that which is familiar to physicists as “close packing.” 
It is a very simple case of what Lord Kelvin used to call 
“a problem in tactics.” It is a mere question of the rigidity 
of the system, of the freedom of movement on the part of 
its constituent cells, whether or at what stage this tendency 
to slip into the closest propinquity, or position of minimum 
potential, will be found to manifest itself. 

However the hexactinellid spicules be arranged (and this is 

* See, for instance, the figures of the segmenting egg of Synapta (after Selenka), 
in Korschelt and Heider’s Vergleichende Entwicklungsgeschichte (Allgem. Th., 3*¢ 
Lief.), p. 19, 1909. On the spiral type of segmentation as a secondary derivative, 
due to mechanical causes, of the “radial” type of segmentation, see E. B. Wilson, 


Cell-lineage of Nereis, Journ. of Morphology, vi, p. 450, 1892. + 
+ Korschelt and Heider, p. 16. 


454 ON CONCRETIONS, SPICULES, ETC. [CH. 


not at all easy to determine) in relation to the tissues and chambers 
of the sponge, it is at least clear that, whether they be separate 
or be fused together (as often happens) in a composite skeleton, 
they effect a symmetrical partitioning of space according to the 
cubical system, in contrast to that closer packing which is repre- 
sented and effected by the tetrahedral system*. 


This question of the origin and causation of the forms of 
sponge-spicules, with which we have now briefly dealt, is all the 
more important and all the more interesting because it has been 
discussed time and again, from points of view which are charac- 
teristic of very different schools of thought in biology. Haeckel 
found in the form of the sponge-spicule a typical illustration of 
his theory of “bio-crystallisation’; he considered that these 
“biocrystals”’ represented “something midway—ein Muittelding— 
between an inorganic crystal and an organic secretion”; that 
there was a “compromise between the crystallising efforts of the 
calcium carbonate and the formative activity of the fused cells 
of the syncytium”; and that the semi-crystalline secretions of 
calcium carbonate “were utilised by natural selection as ‘spicules’ 
for building up a skeleton, and afterwards, by the interaction of 
adaptation and heredity, became modified in form and differen- 
tiated in a vast variety of ways in the struggle for existence f.” 
What Haeckel precisely signified by these words is not clear to me. 

F. E. Schultze, perceiving that identical forms of spicule were 
developed whether the material were crystalline or non-crystalline, 
abandoned all theories based upon crystallisation; he simply saw 
in the form and arrangement of the spicules something which 
was “best fitted” for its purpose, that is to say for the support 
and strengthening of the porous walls of the sponge, and found 
clear evidence of “utility” in the specific structure of these 
skeletal elements. 


* Chall. Rep. Hexactinellida, pls. xvi, liii, Ixxvi, 1xxxviii. 

+ “Hierbei nahm der kohlensaure Kalk eine halb-krystallinische Beschaffen- 
heit an, und gestaltete sich unter Aufnahme von Krystallwasser und in Verbindung 
mit einer geringen Quantitat von organischer Substanz zu jenen individuellen, 
festen K6rpern, welche durch die natiirliche Ziichtung als Spicula zur Skeletbildung 
beniitzt, und spiterhin durch die Wechselwirkung von Anpassung und Vererbung 
im Kampfe ums Dasein auf das Vielfaltigste umgebildet und differenziert wurden.” 
Die Kalkschwdémme, 1, p. 377, 1872; cf. also pp. 482, 483. 


1x] OF THE SKELETON OF SPONGES ADS 


Sollas and Dreyer, as we have seen, introduced in various 
ways the conception of physical causation,—as indeed Haeckel 
himself had done in regard to one particular, when he supposed 
the position of the spicules to be due to the constant passage of 
the water-currents. Though even here, by the way, if I under- 
stand Haeckel aright, he was thinking not merely of a direct or im- 
mediate physical causation, but of one manifesting itself through 
the agency of natural selection*. Sollas laid stress upon the “ path 
of least resistance” as determining the direction of growth; 
while Dreyer dealt in greater detail with the various tensions 
and pressures to which the growing spicule was exposed, amid 
the alveolar or vesicular structure which was represented alike 
by the chambers of the sponge, by the reticulum of constituent 
cells, or by the minute structure of the intracellular protoplasm. 
But neither of these writers, so far as I can discover, was inclined 
to doubt for a moment the received canon of biology, which sees 
in such structures as these the characteristics of true organic 
species, and the indications of an hereditary affinity by which 
blood-relationship and the succession of evolutionary descent 
throughout geologic time can be ultimately deduced. 

Lastly, Minchin, in a well-known papert, took sides with 
Schultze, and gave reasons for dissenting from such mechanical 
theories as those of Sollas and of Dreyer. For example, after 
pointing out that all protoplasm contains a number of “ granules” 
or microsomes, contained in the alveolar framework and lodged 
at the nodes of the reticulum, he argued that these also ought to 
acquire a form such as the'spicules possess, if it were the case that 
these latter owed their form to their very similar or identical 
position. “If vesicular tension cannot in any other instance cause 
the granules at the nodes to assume a tetraxon form, why should 
it do so for the sclerites?”” In all probability the answer to this 
question is not far to seek. If the force which the “mechanical” 
hypothesis has in view were simply that of mechanical pressure, 

* Op. cit. p. 483. ‘* Die geordnete, oft so sehr regelmiassige und zierliche Zusam- 
mensetzung des Skeletsystems ist zum gréssten Theile unmittelbares Product 
der Wasserstr6mung; die characteristische Lagerung der Spicula ist von der 
constanten Richtung des Wasserstroms hervorgebracht; zum kleinsten Theile ist 


sie die Folge von Anpassungen an untergeordnete aiussere Existenzbedingungen.” 
+ Materials for a Monograph of the Ascones, Q. J. M. S. xi. pp. 469-587, 1898. 


456 ON CONCRETIONS, SPICULES, ETC. [cH. 


as between solid bodies, then indeed we should expect that any 
substances whatsoever, lying between the impinging spheres, 
would tend (unless they were infinitely hard) to assume the 
quadriradiate or “tetraxon” form; but this conclusion does not 
follow at all, in so far as it is to surface-energy that we ascribe the 
phenomenon. Here the specific nature of the substances involved 
makes all the difference. We cannot argue from one substance 
to another; adsorptive attraction shews its effect on one and not 
on another; and we have not the least reason to be surprised if — 
we find that the little granules of protoplasmic material, which 
as they lie bathed in the more fluid protoplasm have (presumably, 
and as their shape indicates) a strong surface-tension of their 
own, behave towards the adjacent vesicles in a very ‘different 
fashion to the incipient aggregations of calcareous or siliceous 
matter ina colloid medium. ‘‘The ontogeny of the spicules,” says 
Professor Minchin, “ points clearly to their regular form being a 
phylogenetic adaptation, which has become fixed and handed on by 
heredity, appearing in the ontogeny as a prophetic adaptation.” 
And again, “The forms of the spicules are the result of adaptation 
to the requirements of the sponge as a whole, produced by the 
action of natural selection wpon variation im every direction.” It 
would scarcely be possible to illustrate more briefly and more 
cogently than by these few words (or the similar words of Haeckel 
quoted on p. 454), the fundamental difference between the 
Darwinian conception of the causation and determination of 
Form, and that which is characteristic of the physical sciences. 


If I have dealt comparatively briefly with the imorganic 
skeleton of sponges, in spite of the obvious importance of this 
-part of our subject from the physical or mechanical point of view, 
it has been owing to several reasons. In the first place, though 
the general trend of the phenomena is clear, it must be at once 
admitted that many points are obscure, and could only be discussed 
at the cost of a long argument. In the second place, the physical 
theory is (as I have shewn) in manifest conflict with the accounts 
given by various embryologists of the development of the spicules, 
and of the current biological theories which their descriptions 
embody; it is beyond our scope to deal with such descriptions 


Ix| OF THE RADIOLARIAN SKELETON 457 


in detail. Lastly, we find ourselves able to illustrate the same 
physical principles with greater clearness and greater certitude in 
another group of animals, namely the Radiolaria. In our descrip- 
tion of the skeletons occurring within this group we shall by no 
means abandon the preliminary classification of microscopic 
skeletons which we have laid down; but we shall have occasion 
to blend with it the consideration of certain other more or less 
correlated phenomena. 

The group of microscopic organisms known as the Radiolaria 
is extraordinarily rich in diverse forms, or “species.” I do not 
know how many of such species have been described and defined 
by naturalists, but some thirty years ago the number was said 
to be over four thousand, arranged in more than seven hundred 
genera*. Of late years there has been a tendency to reduce the 
number, it being found that some of the earlier species and even 
genera are but growth-stages of one and the same form, sometimes 
mere fragments or “ fission-products”? common to several species, 
or sometimes forms so similar and so interconnected by inter- 
mediate forms that the naturalist denominates them not “species” 
but “varieties.” It has to be admitted, in short, that the con- 
ception of species among the Radiolaria has not hitherto been, 
and is not yet, on the same footing as that among most other 
groups of animals. But apart from the extraordinary multiplicity 
of forms among the Radiolaria, there are certain other features 
in this multiplicity which arrest our attention. For instance, 
the distribution of species in space is curious and vague; many 
species are found all over the world, or at least every here and 
there, with no evidence of specific limitations of geographical 
habitat; others occur in the neighbourhood of the two poles; 
some are confined to warm and others to cold currents of the 
ocean. In time also their distribution is not less vague: so much 
so that it has been asserted of them that “from the Cambrian 
age downwards, the families and even genera appear identical 
with those now living.” Lastly, except perhaps in the case of 
a few large “colonial forms,” we seldom if ever find, as is usual 


* Haeckel, in his Challenger Monograph, p. clxxxviii (1887) estimated the 
number of known forms at 4314 species, included in 739 genera. Of these, 3508 
species were described for the first time in that work. 


458 ON CONCRETIONS, SPICULES, ETC. [CH. 


in most animals, a local predominance of one particular species. 
On the contrary, in a little pinch of deep-sea mud or of some fossil 
“Radiolarian earth,’ we shall probably find scores, and it may be 
even hundreds, of different forms. Moreover, the radiolarian 
skeletons are of quite extraordinary delicacy and complexity, in 
spite of their minuteness and the comparative simplicity of the 
“unicellular” organisms within which they grow; and these 
complex conformations have a wonderful and unusual.appearance 
of geometric regularity. All these general considerations seem 
such as to prepare us for the special need of some physical 
hypothesis of causation. The little skeletal fabrics remind us of 
such objects as snow-crystals (themselves almost endless in their 
diversity), rather than of a collection of distinct animals, con- 
structed in apparent accordance with functional needs, and dis- 
tributed in accordance with their fitness for particular situations. 
Nevertheless great efforts have been made of recent years to 
attach “a biological meaning” to these elaborate structures ; 
and “to justify the hope that in time the utilitarian character 
(of the skeleton} will be more completely recognised*.” 

In the majority of cases, the skeleton of the Radiolaria is 
composed, like that of so many sponges, of silica; in one large 
family, the Acantharia (and perhaps in some others), it is composed, 
in great part at least, of a very unusual constituent, namely 
strontium sulphatey. There is no fundamental or important — 
morphological character in which the shells formed of these two 
constituents differ from one another; and in no case can the 
chemical properties of these inorganic materials be said to influence 
the form of the complex skeleton or shell, save only in this general 
way that, by their rigidity and toughness, they may give rise to 
a fabric far more delicate and slender than we find developed 
among calcareous organisms. 

A slight exception to this rule is found in the presence of true 
crystals, which occur within the central capsules of certain Radio- 


* Cf. Gamble, Radiolaria (Lankester’s Treatise on Zoology), vol. 1, p. 131, 1909. 
Cf. also papers by V. Hacker, in Jen. Zeitschr. xxx1x, p. 581, 1905, Z. f. wiss. 
Zool. LXxXxt, p. 336, 1905, Arch. f. Protistenkunde, 1x, p. 139, 1907, ete. 

+ Biitschli, Ueber die chemische Natur der Skeletsubstanz der Acantharia, 
Zool. Anz. ¥Xx, p. 784, 1906. 


1x} OF THE RADIOLARIAN SKELETON 459 


laria, for instance the genus Collosphaera*. Johannes Miiller 
(whose knowledge and insight never fail to astonish us) remarked 
that these were identical in form with crystals of celestine, a 
sulphate of strontium and barium; and Biitschli’s discovery of 
sulphates of strontium and of barium in kindred forms render it 
all but certain that they are actually true crystals of celestine f. 
In its typical form, the Radiolarian body consists of a spherical 
mass of protoplasm, around which, and separated from it by some 
sort of porous “capsule,” les a frothy mass, composed of proto- 
plasm honeycombed into a multitude of alveoli or vacuoles, filled 
with a fluid which can scarcely differ much from sea-water tf. 
According to their surface-tension conditions, these vacuoles may 
appear more or less isolated and spherical, or joining together in 
a “froth” of polygonal cells; and in the latter, which is the 
commoner condition, the cells tend to be of equal size, and the 
resulting polygonal meshwork beautifully regular. In many cases, 
a large number of such simple individual organisms are associated 
together, forming a floating colony, and it is highly probable that 
manv other forms, with whose scattered skeletons we are alone 
acquainted, had in life formed part likewise of a colonial organism. 
In contradistinction to the sponges, in which the skeleton 
always begins as a loose mass of isolated spicules, which only in 
a few exceptional cases (such as Kuplectella and Farrea) fuse into 
a continuous network, the characteristic feature of the Radiolarians 
hes in the possession of a continuous skeleton, in the form of a 
netted mesh or perforated lacework, sometimes however replaced . 
by and often associated with minute independent spicules. Before 
we proceed to treat of the more complex skeletons, we may begin, 
then, by dealing with these comparatively simple cases where 
either the entire skeleton or a considerable part of it is represented, 
not by a continuous fabric, but by a quantity of loose, separate 
spicules, or aciculae, which seem, like the spicules of Aleyonium, 


* For figures of these crystals see Brandt, F. u. Fl. d. Golfes von Neapel, X11. 
Radiolaria, 1885, pl. v. Cf. J. Miller, Ueber die Thalassicollen, etc. Abh. K. 
Akad. Wiss. Berlin, 1858. 

+ Celestine, or celestite, is StSO, with some BaO replacing SrO. 

{ With the colloid chemists, we may adopt (as Rhumbler has done) the terms 
spumoid or emulsoid to denote an agglomeration of fluid-filled vesicles, restricting 
the name froth to such vesicles when filled with air or some other gas. 


460 ON CONCRETIONS, SPICULES, (cH. 


to be developed as free and isolated formations or deposits, 
precipitated in the colloid matrix, with no relation of form to 
the cellular or vesicular boundaries. These simple acicular spicules 
occupy a definite position in the organism. Sometimes, as for 
instance among the fresh-water Heliozoa (e.g. Raphidiophrys), they 
lie on the outer surface of the organism, and not infrequently 
(when the spicules are few in number) they tend to collect round 
the bases of the pseudopodia, or around the large radiating 
spicules, or axial rays, in the cases where these latter are present. 
When the spicules are thus localised around some prominent centre, 
they tend to take up a position of symmetry in regard toit; instead 
of forming a tangled or felted layer, they come to lie side by side, 
in a radiating cluster round the focus. In other cases (as for 
instance in the well-known Radiolarian Aulacantha scolymantha) 
the felted layer of aciculae lies at some depth below the surface, 
forming a sphere concentric with the entire spherical organism. 
In either case, whether the layer of spicules be deep or be super- 
ficial, it tends to mark a “surface of discontinuity,” a meeting 
place between two distinct layers of protoplasm or between the 
protoplasm and the water around; and it is obvious that, in either 
case, there are manifestations of surface-energy at the boundary, 
which cause the spicules to be retained there, and to take up their 
position in its plane. The case is somewhat, though not directly, 
analogous to that of a cirrus cloud, 
which marks the place of a surface 
! of discontinuity in a stratified at- 
; mosphere. 

We have, then, to enquire what 
! are the conditions which shall, apart 


ae from gravity, confine an extraneous 
\e body to a surface-film; and we may 
ASG do this very simply, by considering 


S 


\ the surface-energy of the entire 

; { system. In Fig. 218 we have two 
\ \ fluids in contact with one another 
iis, 318 (let us call them water and proto- 


plasm), and a body (b) which may 
be immersed in either, or may be restricted to the boundary 


P 


Ix] AND SPICULAR SKELETONS 461 


between. We have here three possible “interfacial contacts,” 
* each with its own specific surface-energy, per unit of surface 
area: namely, that between our particle and the water (let us 
call it a), that between the particle and the protoplasm (8), and 
that between water and protoplasm (y). When the body lies 
in the boundary of the two fluids, let us say half in one 
and half in the other, the surface-energies concerned are 
equivalent to (S/2)a + (S/2)8; but we must also remember that, 
by the presence of the particle, a small portion (equal to its 
sectional area s) of the original contact-surface between water 
and protoplasm has been obliterated, and with it a proportionate 
quantity of energy, equivalent to sy, has been set free. When, 
on the other hand, the body lies entirely within one or other 
fluid, the surface-energies of the system (so far as we are concerned) 
are equivalent to Sa+sy, or SB+ sy, as the case may be. 
According as a be less or greater than f, the particle will have 
a tendency to remain immersed in the water or in the protoplasm ; 
but if (S/2) (a + B) — sy be less than either Sa or Sf, then the 
condition of minimal potential will be found when the particle 
lies, as we have said, in the boundary zone, half in one fluid and 
half in the other; and, if we were to attempt a more general 
solution of the problem, we should evidently have to deal with 
possible conditions of equilibrium under which the necessary 
balance of energies would be attained by the particle rising or 
sinking in the boundary zone, so as to adjust the relative magni- 
tudes of the surface-areas concerned. It is obvious that this 
principle may, in certain cases, help us to explain the position 
even of a radial spicule, which is just a case where the surface of 
the solid spicule is distributed between the fluids with a minimal 
disturbance, or minimal replacement, of the original surface of 
contact between the one fluid and the other. 

In like manner we may provide for the case (a common and 
an important one) where the protoplasm “creeps up” the spicule, 
covering it with a delicate film. In Acanthocystis we have 
yet another special case, where the radial spicules plunge only 
a certain distance into the protoplasm of the cell, being arrested 
at a boundary-surface between an inner and an outer layer of 
cytoplasm; here we have only to assume that there is a tension 


462 ON CONCRETIONS, SPICULES, ETC. [CH. 


at this surface, between the two layers of protoplasm, sufficient 
to balance the tensions which act directly on the spicule*. 

In various Acanthometridae, besides such typical characters 
as the radial symmetry, the concentric layers of protoplasm, and 
the capillary surfaces in which the outer, vacuolated protoplasm 
is festooned upon the projecting radii, we have another curious 
feature. On the surface of the protoplasm where it creeps up 
the sides of the long radial spicules, we find a number of elongated 
bodies, forming in each case one or several little groups, and 
lying neatly arranged in parallel bundles. A Russian naturalist, 
Schewiakoff, whose views have been accepted in the text-books, 
tells us that these are muscular structures, serving to raise or 
lower the conical masses of protoplasm about the radial spicules, 
which latter serve as so many “tent-poles” or masts, on which 
the protoplasmic membranes are hoisted up; and the little 
elongated bodies are dignified with various names, such as 
“myonemes”’ or “myophrises,” in allusion to their supposed 
muscular naturet. This explanation is by no means convincing. 
To begin with, we have precisely similar festoons of protoplasm 
in a multitude of other cases where the “myonemes” are lacking; 
from their minute size (-006—-012 mm.) and the amount of con- 
traction they are said to be capable of, the myonemes can hardly 
be very efficient instruments of traction; and further, for them 
to act (as is alleged) for a specific purpose, namely the “hydrostatic 
regulation” of the organism giving it power to sink or to swim, 
would seem to imply a mechanism of action and of coordination 
which is difficult to conceive in these minute and simple organisms. 
The fact is (as it seems to me), that the whole method of explana- 
tion is unnecessary. Just as the supposed “hauling up” of the 
protoplasmic festoons is at once explained by capillary phenomena, 
so also, in all probability, is the position and arrangement of 
the little elongated bodies. Whatever the actual nature of these 
bodies may be, whether they are truly portions of differentiated 
protoplasm, or whether they are foreign bodies or spicular 
structures (as bodies occupying a similar position in other cases 
undoubtedly are), we can explain their situation on the surface 


* Cf. Koltzoff, Zur Frage der Zellgestalt, Anat. Anzeiger, xxi, p. 190, 1912. 
+ Mém. del Acad. des Sci., St. Pétersbourg, x11, Nr. 10, 1902. 


Ix] OF AGGLUTINATED SKELETONS 463 


of the protoplasm, and their arrangement around the radial 
spicules, all on the principles of surface-tension *. 

This last case is not of the simplest; and I do not forget that 
my explanation of it, which is wholly theoretical, implies a doubt 
of Schewiakofi’s statements, which are founded on direct personal 
observation. This I am none too willing to do; but whether it 
be justly done in this case or not, I hold that it is in principle 
justifiable to look with great suspicion upon a number of kindred 
statements where it is obvious that the observer has left out of 
account the purely physical aspect of the phenomenon, and all 
the opportunities of simple explanation which the consideration 
of that aspect might afford. 


Whether it be wholly applicable to this particular and complex 
case or no, our general theorem of the localisation and arrestment 
of solid particles in a surface-film is of very great biological 
importance; for on it depends the power displayed by many 
little naked protoplasmic organisms of covering themselves with 
an “agglutinated” shell. Sometimes, as in Difflugia, Astrorhiza 
(Fig. 219) and others, this covering consists of sand-grains picked 
up from the surrounding medium, and sometimes, on the other 
hand, as in Quadrula, it consists of solid particles which are said 
to arise, as inorganic deposits or concretions, within the protoplasm 
itself, and which find their way outwards to a position of equilibrium 
in the surface-layer; and in both cases, the mutual capillary 
attractions between the particles, confined to the boundary-layer 
but enjoying a certain measure of freedom therein, tends to the 
_ orderly arrangement of the particles one with another, and even 
to the appearance of a regular “pattern” as the result of this 
arrangement. 

The “picking up” by the protoplasmic organism of a solid 
particle with which “to build its house” (for it is hard to avoid 
this customary use of anthropomorphic figures of speech, misleading 
though they be), isa physical phenomenon kindred to that by which 
an Amoeba “swallows” a particle of food. This latter process 
has been reproduced or imitated in various pretty experimental 


* The manner in which the minute spicules of Raphidophrys arrange themselves 
round the bases of the pseudopodial rays is a similar phenomenon. 


Fig, 219. Arenaceous Foraminifera; Astrorhiza limicola and arenaria. 
(From Brady’s Challenger Monograph.) 


CH. 1x] ON CONCRETIONS, SPICULES, ETC. 465 


ways. For instance, Rhumbler has shewn that if a thread of 
glass be covered with shellac and brought near a drop of 
chloroform suspended in water, the drop takes in the spicule, 
robs it of its shellac covering, and then passes it out again*. 
It is all a question of relative surface-energies, leading to different 
degrees of “adhesion”? between the chloroform and the glass or 
its covering. Thus it is that the Amoeba takes in the diatom, 
dissolves off its proteid covering, and casts out the shell. 
Furthermore, as the whole phenomenon depends on a distribu- 
tion of surface-energy, the amount of which is specific to certain 
particular substances in contact with one another, we have no 
difficulty in understanding the selective action, which is very often 
a conspicuous feature in the phenomenon. Just as some caddis- 
worms make their houses of twigs, and others of shells and again 
others of stones, so some Rhizopods construct their agglutinated 
“test” out of stray sponge-spicules, or frustules of diatoms, or 
again of tiny mud particles or of larger grains of sand. In all 
these cases, we have apparently to deal with differences in specific 


* Rhumbler, Physikalische Analyse von Lebenserscheinungen der Zelle, Arch, 
f. Entw. Mech. vu, p. 103, 1898. 

+ The whole phenomenon is described by biologists as a “surprising exhibition 
of constructive and selective activity,” and is ascribed, in varying phraseology, to 
intelligence, skill, purpose, psychical activity, or “microscopic mentality”: that is 
to say, to Galen’s rexvixh pious, or “ artistic creativeness ” (cf. Brock’s Galen, 1916, 
p- xxix). Cf. Carpenter, Mental Physiology, 1874, p. 41; Norman, Architectural 
achievements of Little Masons, etc., Ann. Mag. Nat. Hist. (5), 1, p. 284, 1878; Heron- 
Allen, Contributions...to the Study of the Foraminifera, Phil. Trans. (B), Ccvt, 
pp. 227-279, 1915; Theory and Phenomena of Purpose and Intelligence exhibited by 
the Protozoa, as illustrated by selection and behaviour in the Foraminifera, Journ. h. 
Microscop. Soc. pp. 547-557, 1915; ibid., pp. 137-140, 1916. Prof. J. A. Thomson 
(New Statesman, Oct. 23, 1915) describes a certain little foraminifer, whose proto- 
plasmic body is overlaid by a crust of sponge-spicules, as ‘‘a psycho-physical 
individuality whose experiments in self-expression include a masterly treatment of 
sponge-spicules, and illustrate that organic skill which came before the dawn of Art.” 
Sir Ray Lankester finds it “not difficult to conceive of the existence of a mechanism 
in the protoplasm of the Protozoa which selects and rejects building-material, and 
determines the shapes of the structures built, comparable to that mechanism which 
is assumed to exist in the nervous system of insects and other animals which 
‘automatically’ go through wonderfully elaborate series of complicated actions.” 
And he agrees with “Darwin and others [who] have attributed the building up of 
these inherited mechanisms to the age-long action of Natural Selection, and the 
survival of those individuals possessing qualities or ‘tricks’ of life-saving value,” 
J. R. Microsc. Soc. April, 1916, p. 136. 


T. G. 30 


466 ON CONCRETIONS, SPICULES, [CH. 


surface-energies, and also doubtless with differences in the total 
available amount of surface-energy in relation to gravity or other 
extraneous forces. In my early student days, Wyville Thomson 
used to tell us that certain deep-sea “ Difflugias,”’ after constructing 
a shell out of particles of the black volcanic sand common in parts 
of the North Atlantic, finished it off with “a clean white collar” 
of little grains of quartz. Hven this phenomenon may be accounted 
for on surface-tension principles, if we assume that the surface- 
energy ratios have tended to change, either with the growth of 
the protoplasm or by reason of external variation of temperature 
or the like; and we are by no means obliged to attribute the 
phenomenon to a manifestation of volition, or taste, or aesthetic 
skill, on the part of the microscopic organism. Nor, when certain 
Radiolaria tend more than others to attract into their own sub- 
stance diatoms and such-like foreign bodies, is it scientifically 
correct to speak, as some text-books do, of species ‘‘in which 
diatom selection has become a regular habit.” To do so is an 
exaggerated misuse of anthropomorphic phraseology. 

The formation of an “agglutinated” shell is thus seen to be 
a purely physical phenomenon, and indeed a special case of a 
more general physical phenomenon which has many other 
important consequences in biology. For the shell to assume the 
solid and permanent character which it acquires, for instance, in 
Difflugia, we have only to make the additional assumption that 
some small quantities of a cementing substance are secreted by 
the animal, and that this substance flows or creeps by capillary 
attraction between all the interstices of the little quartz grains, 
and ends by binding them all firmly together. Rhumbler* has 
shewn us how these agglutinated tests, of spicules or of sand- 
grains, can be precisely imitated, and how they are formed with 
greater or less ease, and greater or less rapidity, according to the 
nature of the materials employed, that is to say, according to 
the specific surface-tensions which are involved. For instance if 
we mix up a little powdered glass with chloroform, and set a drop 
of the mixture in water, the glass particles gather neatly round 
the surface of the drop so quickly that the eye cannot follow the 


* Rhumbler, Das Protoplasma als physikalisches System, Jena, p. 591, 1914; 
also in Arch. f. Entwickelungsmech. vu, pp. 279-335, 1898. 


Ix] AND SPICULAR SKELETONS 467 


operation. Ifwe perform the same experiment with oil and fine sand, 
dropped into 70 per cent. alcohol, a still more beautiful artificial 
Rhizopod shell is formed, but it takes some three hours to do. 

It is curious that, just at the very time when Rhumbler was 
thus demonstrating the purely physical nature of the Difflu- 
gian shell, Verworn was studying the same and kindred organisms 
from the older standpoint of an incipient psychology*. But, as 
Rhumbler himself admits, Verworn was very careful not to over- 
estimate the apparent signs of volition, or selective choice, in the 
little organism’s use of the material of its dwelling. 


This long parenthesis has led us away, for the time being, 
from the subject of the Radiolarian skeleton, and to that subject 
we must now return. Leaving aside, then, the loose and scattered 
spicules, which we have sufficiently discussed, the more perfect 
Radiolarian skeletons consist of a continuous and regular structure ; 
and the siliceous (or other inorganic) material of which this frame- 
work is composed tends to be deposited in one or other of two 
ways or in both combined: (1) in the form of long spicular axes, 
usually conjoined at, or emanating from, the centre of the proto- 
‘plasmic body, and forming a symmetric radial system; (2) in the 
form of a crust, developed in various ways, either on the outer 
surface of the organism or in relation to the various internal 
surfaces which separate its concentric layers or its component 
vesicles. Not unfrequently, this superficial skeleton comes to 
constitute a spherical shell, or a system of concentric or otherwise 
associated spheres. 

We have already learned that a great part of the body of the 
Radiolarian, and especially that outer portion to which Haeckel 
has given the name of the “calymma,” is built up of a great mass 
of “vesicles,” forming a sort of stiff froth, and equivalent in the 
physical sense (though not necessarily in the biological sense) to 
“cells,” inasmuch as the little vesicles have their own well-defined 
boundaries, and their own surface phenomena. In short, all that 
we have said of cell-surfaces, and cell conformations, in our 
discussion of cells and of tissues, will apply in like manner, and 
under appropriate conditions, to these. In certain cases, even in 

* Verworn, Psycho-physiologische Protisten-Studien, Jena, 1889 (219 pp.). 

30—2 


468 ON CONCRETIONS, SPICULES, ETC. [CH. 


so common and simple a one as the vacuolated substance of an 
Actinosphaerium, we may see a very close resemblance, or formal 
analogy, to an ordinary cellular or “ parenchymatous”’ tissue, in the 
close-packed arrangement and consequent configuration of these 
vesicles, and even at times in a slight membranous hardening of 
their walls. Leidy has figured * 
some curious little bodies, like 
small masses of consolidated 
froth, which seem to be nothing 
else than the dead and empty 
husks, or filmy skeletons, of 
Actinosphaerium. And Carnoy7{ 
has demonstrated in certain 
cell-nuclei an all but precisely 
similar framework, of extreme 
delicacy and minuteness, as the 
result of partial solidification 
of interstitial matter in a close- 
packed system of alveoli (Fig. 
220). . 
Fig. 220. ‘Reticulum plasmatique.” Let us now suppose that, 
(After Carnoy.) : : : : 
in our Radiolarian, the outer 
surface of the animal is covered by a layer of froth-like vesicles, 
uniform or nearly so in size. We know that their tensions will 
tend to conform them into a “honeycomb,” or regular meshwork 
of hexagons, and that the free end of each hexagonal prism will 
be a little spherical cap. Suppose now that it be at the outer 
surface of the protoplasm (that namely which is in contact with 
the surrounding sea-water), that the siliceous particles have a 
tendency to be secreted or adsorbed; it will at once follow that 
they will show a tendency to aggregate in the grooves which 
separate the vesicles, and the result will be the development of 
a most delicate sphere composed of tiny rods arranged in a regular 
hexagonal network (e.g. Aulonia). Such a conformation is 


= 


(i 
“Y/ 
Y 


ASS 


* Leidy, J., Fresh-water Rhizopods of N. America, 1879, p. 262, pl. xli, 
figs. 11, 12. 

+ Carnoy, Biologie Cellulaire, p. 244, fig. 108; cf. Dreyer, op. cit. 1892, 
fig. 185. 


, 


1x] OF THE RADIOLARIAN SKELETON 469 


extremely common, and among its many variants may be found 
cases in which (e.g. Actonomma), the vesicles have been less 


Fig. 221. Aulonia hexagona, Hkl. 


Fig 222. Actinomma arcadophorum, Hkl. 


regular in size, and some in which the hexagonal meshwork has 
been developed not only on one outer surface, but at successive 


470 ON CONCRETIONS, SPICULES, ETC. [CH. 


surfaces, producing a system of concentric spheres. If the siliceous 
material be not limited to the linear junctions of the cells, but 
spread over a portion of the outer spherical surfaces or caps, then 
we shall have the condition represented in Fig. 223 (Hthmosphaera), 
where the shell appears perforated by circular instead of hexagonal 
apertures, and the circular pores are set on slight spheroidal 
eminences; and, interconnected with such types as this, we have 
others in which the accumulating pellicles of skeletal matter have 
extended from the edges into the substance of the boundary walls 


Fig. 223. Hthmosphaera conosiphonia, Fig. 224. Portions of shells 
Akl. of two “species” of 
Cenosphaera : upper 


figure, C. favosa, lower, 
C. vesparia, Hkl. 


and have so produced a system of films, normal to the surface of 
the sphere, constituting a very perfect honeycomb, as in Ceno- 
sphaera favosa and vesparia*. : 

In one or two very simple forms, such as the fresh-water 
Clathrulina, just such a spherical perforated shell is produced out 
of some organic, acanthin-like substance; and in some examples 
of Clathrulina the chitinous lattice-work of the shell is just as 


* Tn all these latter cases we recognise a relation to, or extension of, the principle 
of Plateau’s bourrelet, or van der Mensbrugghe’s masse annulaire, of which we have 
already spoken (p. 297). 


Ix] OF THE RADIOLARIAN SKELETON 471 


regular and delicate, with the meshes just as beautifully hexagonal, 
as in the siliceous shells of the oceanic Radiolaria. This is only 
another proof (if proof be needed) that the peculiar conformation 
of these little skeletons is not due to the material of which they 
are composed, but to the moulding of that material upon an under- 
lying vesicular structure. 

Let us next suppose that, upon some such lattice-work as has 
just been described, another and external layer of cells or vesicles 
is developed, and that instead of (or perhaps only in addition to) 
a second hexagonal lattice-work, which might develop concen- 


Fig. 225. Aulastrum triceros, Hkl. 


trically to the first in the boundary-furrows of this new layer of 
cells, the siliceous matter now tends to be deposited radially, 
or normally to the surface of the sphere, just in the lines where 
the external layer of vesicles meet one another, three by three. 
The result will be that, when the vesicles themselves are removed, 
a series of radiating spicules will be revealed, directed outwards 
from each of the angles of the original hexagon; as is seen 
in Fig. 225. And it may further happen that these radiating 
skeletal rods are continued at their distal ends into divergent 
rays, forming a triple fork, and corresponding (after a fashion 


472 ON CONCRETIONS, SPICULES, ETC. [CH. 


which we have already described as occurring in certain sponge- 
spicules) to the three superficial furrows between the adjacent 
cells. This last is, as it were, an intermediate stage between the 
simple rods and the complete formation of another concentric 
sphere of latticed hexagons. Another possible case is when the 
large and uniform vesicles of the outer protoplasm are mixed 
with, or replaced by, much smaller vesicles, piled on one another 
in more or less concentric layers; in this case the radiating rods 


SAA a) 


A 


i 
¥en 
INS Pets 
SEN aa Se: 


A 
DAA 


Sy 


VON Oe 


38 
ae 
hat 
Bie 
al 


r 


ie 


UA 


will no longer be straight, but will be bent into a zig-zag pattern, 
with angles in three vertical planes, corresponding to the suc- 
cessive contacts of the groups of cells around the axis (Fig. 226). 


Among a certain group called the Nassellaria, we find geome- 
trical forms of peculiar simplicity and beauty,—such for instance 
as that which I have represented in Fig. 227. It is obvious at 
a glance that this is such a skeleton as may have been formed 

} 


1X] OF THE NASSELLARIAN SKELETON 473 


(1 think we may go so far as to say must have been formed) at 
the interfaces of a little tetrahedral group of cells, the four equal 
cells o the tetrahedron being in this particular case supplemented 
by a little one in the centre of the system. We see, precisely as 
in the internal boundary-system of an artificial group of four 
soap-bubbles, the plane surfaces of contact,- six in number; the 
relation to one another of each triple set of interfacial planes, 
meeting one another at equal angles of 120°; and finally the 
relation of the four lines or edges of triple contact, which tend 
(but for the little central vesicle) to meet at co-equal solid angles 
in the centre of the system, all as we have described on p. 318. 
In short, each triple-walled re-entrant angle of the little shell has 
essentially the configuration (or a part thereof) of what we have 
called a “Maraldi pyramid” in our account of the architecture of 
the honeycomb, on p. 329*. 

There are still two or three remarkable or peculiar features in 
this all but mathematically perfect shell, and they are in part easy 
and in part they seem more difficult of interpretation. 

We notice that the amount of solid matter deposited in the 
plane interfacial boundaries is greatly increased at the outer 
margin of each boundary wall, where it merges or coincides with 
the superficial furrow which separates the free, spherical surfaces 
of the bubbles from one another; and we may sometimes find that, 
along these edges, the skeleton remains complete and strong, 
while it shows signs of imperfect development or of breaking 
away over great part of the rest of the interfacial surfaces. In 
this there is nothing anomalous, for we have already recognised 
that it is at the edges or margins of the interfacial partition-walls 
that the manifestation of surface-energy will tend to reach its 
maximum. And just as we have seen that, in certain of our 
“multicellular” spherical Radiolarians, it is at the superficial 


* Apart from the fact that the apex of each pyramid is interrupted, or truncated, 
by the presence of the little central cell, it is also possible that the solid angles 
are not precisely equivalent to those of Maraldi’s pyramids, owing to the fact that 
there is a certain amount of distortion, or axial asymmetry, in the Nassellarian 
system. In other words (to judge from Haeckel’s figures), the tetrahedral symmetry 
in Nassellaria is not absolutely regular, but has a main axis about which three of 
the trihedral pyramids are symmetrical, the fourth having its solid angle somewhat 
diminished. 


474 ON CONCRETIONS, SPICULES, ETC. [CH. 


edges or borders of the partitions, and here only, that skeletal 
formation occurs (giving rise to the 
netted shell with its hexagonal meshes 
of Fig. 221), so also at times, in the 
case of such little aggregates of cells 
or vesicles as the four-celled system 
of Callimitra, it may happen that 
about the external boundary-lines, 
and not in the interior boundary- 
planes, the whole of the skeletal 
matter is aggregated. In Fig. 228 we 
see a curious little skeletal struc- 
ture or complex spicule, whose conformation is easily accounted 
for after this fashion. Little spicules such as this form 
isolated portions of the skeleton in the genus Dictyocha, and 
occur scattered over the spherical surface of the organism 


Fig. 228. An isolated portion of 
the skeleton of Dictyocha. 


Fig. 229. Dictyocha stapedia, Hkl. 


(Fig. 229). The more or less basket-shaped spicule has evidently 
been developed about a little cluster of four cells or vesicles, 
lying in or on the plane of the surface of the organism, and there- 
fore arranged, not in the tetrahedral form of Callimitra, but in 
the manner in which four contiguous: cells lying side by side 
normally set themselves, like the four cells of a segmenting egg: 
that is to say with an intervening “ polar furrow,” whose ends mark 
the meeting place, at equal angles, of the cells in groups of three. 

The little projecting spokes, or spikes, which are set normally 
to the main basket-work, seem to be incompleted portions of 
a larger basket, or in other words imperfectly formed elements 
corresponding to the interfacial contacts in the surrounding parts 


Ix] OF THE NASSELLARIAN SKELETON 475 


ofthesystem. Similar but more complex formations, all explicable 
as basket-like frameworks developed around a cluster of cells, are 
known in great variety. 

In our Nassellarian itself, and in many other cases where the 
plane interfacial boundary-walls are skeletonised, we see that the 
siliceous matter is not deposited in an even and continuous layer, 
like the waxen walls of a bee’s cell, but constitutes a meshwork 
of fine curvilinear threads; and the curves seem to run, on the 
whole, isogonally, and to form three main series, one approxi- 
mately. parallel to, or concentric with, the outer or free edge of 
the partition, and the other two related severally to its two edges 
of attachment. Sometimes (as may also be seen in our figure), 
the system is still further complicated by a fourth series of linear 
elements, which tend to run radially from the centre of the system 
to the free edge of each partition. As regards the former, their 
arrangement is such as would result if deposition or solidification 
had proceeded in waves, starting independently from each of the 
three boundaries of the little partition-wall; and something of 
this kind is doubtless what has happened. We are reminded at 
once of the wave-like periodicity of the Liesegang phenomenon. 
But apart from this we might conceive of other explanations. 
For instance, the liquid film which originally constitutes the 
partition must easily be thrown into wibrations, and (like the dust 
upon a Chladni’s plate) minute particles of matter in contact with 
the film would tend to take up their position in a symmetrical 
arrangement, in direct relation to the nodal points or lines of the 
vibrating surface *. Some such explanation as this(to my thinking) 
must be invoked to account for the minute and varied and very 
beautiful patterns upon many diatoms, the resemblance of which 
patterns (in certain of their simpler cases) to the Chladni figures 
is sometimes striking and obvious. But the many special pro- 
blems which the diatom skeleton suggests I have not attempted 
to consider. 

The last peculiarity of our Nassellarian hes in an apparent 
departure from what we should at first expect in the way of its 


* Cf. Faraday’s beautiful experiments, On the Moving Groups of Particles 
found on Vibrating Elastic Surfaces, ete.. Phil. Trans. 1831, p. 299; Researches 
in Chem. and Phys. 1859, pp. 314-358. 


476 ON CONCRETIONS, SPICULES, [ox 


external symmetry. Were the system actually composed of four 
spherical vesicles in mutual contact, the outer margin of each of 
the six interfacial planes would obviously be a circular arc; and 
accordingly, at each angle of the tetrahedron, we should expect 
to have a depressed, or re-entrant angle, instead of a prominent 
cusp. This is all doubtless due to some simple balance of tensions, 
whose precise nature and distribution is meanwhile a matter of 
conjecture. But it seems as though an extremely simple explana- 
tion would go a long way, and possibly the whole way, to meet 
this particular case. In our ordinary plane diagram of three cells, — 
or soap-bubbles, in contact, we know (and we have just said) 
that the tensions of the three partitions draw inwards the outer 
walls of the system, till at each point of triple contact (P) we tend 


2 Ue a 
\ Ma 
| we e 
| 
gi \ y / TA 
| 
\ 
S 
%; ee =) 
eat Dal ee 
Fig. 230. 


to get a triradiate, equiangular junction. But if we introduce 
another bubble into the centre of the system (Fig. 230), then, as 
Plateau shewed, the tensions of its walls and those of the three 
partitions by which it is now suspended, again balance one 
another, and the central bubble appears (in plane projection) as 
a curvilinear, equilateral triangle. We have only got to convert 
this plane diagram into that of a tetrahedral solid to obtain almost 
precisely the configuration which we are seeking to explain. 
Now we observe that, so far as our figure of Callimitra informs 
us, this is just the shape of the little bubble which occupies the 
centre of the tetrahedral system in that Radiolarian skeleton. 
And I conceive, accordingly, that the entire organism was not 
limited to the four cells or vesicles (together with the little central 


1x] AND SPICULAR SKELETONS 477 


fifth) which we have hitherto been imagining, but there must have 
been an outer tetrahedral system, enclosing the cells which fabri- 
- cated the skeleton, just as these latter enclosed, and deformed, 
the little bubble in the centre of all. We have only to suppose 
that this hypothetical tetrahedral series, forming the outer layer or 
surface of the whole system, was for some chemico-physical reason 
incapable of secreting at its interfacial contacts a skeletal fabric *. 

In this hypothetical case, the edges of the skeletal system would 
be circular ares, meeting one another at an angle of 120°, or, in the 
solid pyramid, of 109°: and this latter is very nearly the condition 
which our little skeleton actually displays. But we observe in 
Fig. 227 that, in the immediate neighbourhood of the tetrahedral 
angle, the circular ares are slightly drawn out into projecting 
cusps (cf. Fig. 230, B). There is no S-shaped curvature of the 
tetrahedral edges as a whole, but a very slight one, a very slight 
change of curvature; close to the apex. This, I conceive, is 
nothing more than what, in a material system, we are bound to 
have, to represent a “surface of continuity.” It is a phenomenon 
precisely analogous to Plateau’s “bourrelet,’ which we have 
already seen to be a constant feature of all cellular systems, 
rounding off the sharp angular contacts by which (in our more 
elementary treatment) we expect one film to make its junction 
with another f. 


In the foregoing examples of Radiolaria, the symmetry which 
the organism displays would seem to be identical with that 
symmetry of forces which is due to the assemblage of surface- 
tensions in the whole system; this symmetry being displayed, in 
one class of cases, in a complex spherical mass of froth, and in 


* We need not go so far as to suppose that the external layer of cells wholly 
lacked the power of secreting a skeleton. In many of the Nassellariae figured by 
Haeckel (for there are many variant forms or species besides that represented here), 
the skeleton of the partition-walls is very slightly and scantily developed. In 
such a case, if we imagine its few and scanty strands to be broken away, the central 
tetrahedral figure would be set free, and would have all the appearance of a complete 
and independent structure. 

+ The “bourrelet” is not only, as Plateau expresses it, a “surface of continuity,” 
but we also recognise that it tends (so far as material is available for its production) 
to further lessen the free surface-area. On its relation to vapour-pressure and to 
the stability of foam, see: FitzGerald’s interesting note in Nature, Feb. 1, 1894 
(Works, p. 309). 


478 ON CONCRETIONS, SPICULES, ETC. [CH. IX 


another class in a simpler aggregate of a few, otherwise isolated, 
vesicles. But among the vast number of other known Radiolaria, 
there are certain forms (especially among the Phaeodaria and 
Acantharia) which display a still more remarkable symmetry, the 
origin of which is by no means clear, though surface-tension 
doubtless plays a part in its causation. These are cases in which 
(as in some of those already described) the skeleton consists 
(1) of radiating spicular rods, definite in number and position, 
and (2) of interconnecting rods or plates, tangential to the more 
or less spherical body of the organism, whose form becomes, 
accordingly, that of a geometric, polyhedral sohd. It may be 
that there is no mathematical difference, save one of degree, 
between such a hexagonal polyhedron as we have seen in Aula- 
cantha, and those which we are about to describe; but the greater 
regularity, the numerical symmetry, and the apparent simplicity 
of these latter, makes of them a class apart, and suggests problems 
which have not been solved nor even investigated. 

The matter is sufficiently illustrated by the accompanying 
figures, all drawn from Haeckel’s Monograph of the Challenger 
Radiolaria*. In one of these we see a regular octahedron, in 
another a regular, or pentagonal dodecahedron, in a third a regular 
icosahedron. In all cases the figure appears to be perfectly 
symmetrical, though neither the triangular facets of the octahedron 
and icosahedron, nor the pentagonal facets of the dodecahedron, 
are necessarily plane surfaces. In all of these cases, the radial 
spicules correspond to the solid angles of the figure; and they are, 
accordingly, six in number in the octahedron, twenty in the 
dodecahedron, and twelve in the icosahedron. If we add to these 
three figures the regular tetrahedron, which we have had frequent 
occasion to study, and the cube (which is represented, at least 
in outline, in the skeleton of the hexactinellid sponges), we have 
completed the series of the five regular polyhedra known to 
geometers, the Platonic bodies} of the older mathematicians. It 
is at first sight all the more remarkable that we should here meet 


* Of the many thousand figures in the hundred and forty plates of this beautifully 
illustrated book, there is scarcely one which does not depict, now patently, now 
in pregnant suggestion, some subtle and elegant geometrical configuration. 

+ They were known (of course) long before Plato: [Adrwy dé Kal év rovrous 
mubaryopifer. 


1. Circoporus sex- 


Skeletons of various Radiolarians, after Haeckel. 
furcus; 2. C. octahedrus; 3. Circogonia icosahedra; 4. Circospathi, 


5. Circorrhegma dodecahedra. 


Fig. 231. 


, 


Ss novenda 


480 ON CONCRETIONS, SPICULES, ETC. [CH. 


with the whole five regular polyhedra, when we remember that, 
among the vast variety of crystalline forms known among minerals, 
the regular dodecahedron and icosahedron, simple as they are 
from the mathematical point of view, never occur. Not only do 
these latter never occur in Crystallography, but (as is explained 
in text-books of that science) it has been shewn that they cannot 
occur, owing to the fact that their indices (or numbers expressing 
the relation of the faces to the three primary axes) involve an 
irrational quantity: whereas it is a fundamental law of crystallo- 
graphy, involved in the whole theory of space-partitioning, that 
“the indices of any and every face of a crystal are small whole 
numbers*.” At the same time, an imperfect pentagonal dodeca- 
hedron, whose pentagonal sides are non-equilateral, is common 
among crystals. If we may safely judge from Haeckel’s figures, 
the pentagonal dodecahedron of the Radiolarian is perfectly 
regular, and we must presume, accordingly, that it is not brought 
about by principles of space-partitioning similar to those which 
manifest themselves in the phenomenon of crystallisation. It 
will be observed that in all these radiolarian polyhedral shells, 
the surface of each external facet is formed of a minute hexa- 
gonal network, whose probable origin, in relation to a vesicular 
structure, is such as we have already discussed. 

In certain allied Radiolaria (Fig. 232), which, like the dodeca- 
hedral form figured in Fig. 231, 5, have twenty radial spines, these 
latter are commonly described as being arranged in a certain very 
singular way. It is stated that their arrangement may be referred 


* Tf the equation of any plane face of a crystal be written in the form 
ha +ky+lz=1, then h, k, l are the indices of which we are speaking. They are 
the reciprocals of the parameters, or reciprocals of the distances from the origin 
at which the plane meets the several axes. In the case of the regular or pentagonal 
dodecahedron these indices are 2, 1 + sf 5, 0. Kepler described as follows, briefly 
but adequately, the common characteristics of the dodecahedron and icosahedron : 
“Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quin- 
quangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam 
coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura 
perfici non potest sine proportione illa, quam hodierni geometrae divinam appellant” 
(De nive sexangula (1611), Opera, ed. Frisch, vi, p.723). Here Kepler was dealing, 
somewhat after the manner of Sir Thomas Browne, with the mysteries of the 
quincunx, and also of the hexagon; and was seeking for an explanation of the 
mysterious or even mystical beauty of the 5-petalled or 3-petalled flower,—pulehri- 
tudinis aut proprietatis figurae, quae animam harum plantarum characterisavit. 


Ix] OF MULLER’S LAW 481 


to a seriés of five parallel circles on the sphere, corresponding to the 
equator (c), the tropics (b, d) and the polar circles (a, e); and that 
beginning with four equidistant spines in the equator, we have 
alternating whorls of four, radiating outwards from the sphere in 
each of the other parallel zones. This rule was laid down by the 
celebrated Johannes Miiller, and has ever since been used and 
quoted as Miiller’s law. The chief point in this alleged arrange- 
ment which strikes us at first sight as very curious, is that there 
is said to be no spine at either pole; and when we come to examine 
carefully the figure of the organism, we find that the received 


Fig. 232. Dorataspis sp.; diagrammatic. 

description does not do justice to the facts. We see, in the first 
place, from such figures as Figs. 232, 234, that here, unlike our 
former cases, the radial spines issue through the facets (and through 
all the facets) of the polyhedron, instead of through its solid angles ; 
and accordingly, that our twenty spines correspond (not, as before, 
to a dodecahedron) but to some sort of an icosahedron. We see 
in the next place, that this icosahedron is composed of faces, or 
plates, of two different kinds, some hexagonal and some penta- 
gonal; and when we look closer, we discover that the whole 
figure is that of a hexagonal prism, whose twelve solid angles are 
replaced by pentagonal facets. Both hexagons and pentagons 


T._G. 31 


482 ON CONCRETIONS, SPICULES, ETC. [CH. Ix 


appear to be perfectly equilateral, but if we try to construct a 
plane-sided polyhedron of this kind, we soon find that it is 
impossible; for into the angles between the six equatorial hexagons 
those of the six united pentagons will not fit. The figure however 
can be easily constructed if we replace the straight edges (or some 
of them) by curves, and the plane facets by corresponding, slightly 
curved, surfaces. The true symmetry of this figure, then, is 
hexagonal, with a polar axis, produced into two polar spicules; 
with six equatorial spicules, or rays; and with two sets of six 
spicular rays, interposed between the polar axis and the equatorial 
rays, and alternating in position with the latter. 


Miiller’s description was emended by Brandt, and what is now known as 
“ Brandt’s law,”’ viz. that the symmetry consists of two polar rays, and three 
whorls of six each, coincides with the above description so far as the spicular 
axes go: save only that Brandt specifically states that the intermediate 
whorls stand equidistant between the equator and the poles, i.e. in latitude 45°. 
While not far from the truth, this statement is not exact; for according to 
the geometry of the figure, the intermediate cycles obviously stand in a slightly 
higher latitude, but this latitude I have not attempted to determine; for 
the calculation seems to be a little troublesome owing to the curvature of 
the sides of the figure, and the enquiring mathematician will perform it more 
easily than I. Brandt, if I understand him rightly, did not propose his 
“law” as a substitute for Miiller’s law, but as a second law applicable to a few 
particular cases. I on the other hand can find no case to which Miiller’s law 
properly applies. 


If we construct such.a polyhedron, and set it in the position 
of Fig. 232, B, we shall easily see that it is capable of explanation 
(though improperly) in accordance with Miiller’s law; for the 
four equatorial rays of Miiller (c) now correspond to the two polar 
and to two opposite equatorial facets of our polyhedron: the 
four “polar” rays of Miiller (a or e) correspond to two adjacent 
hexagons and two intermediate pentagons of the figure: and 
Miiller’s “tropical” rays (6 or d) are those which emanate from the 
remaining four pentagonal facets, in each half of the figure. In 
some cases, such as Haeckel’s Phatnaspis cristata (Fig. 233), we 
have an ellipsoidal body, from which the spines emerge in the 
order described, but which is not obviously divided by facets. 
In Fig. 234 I have indicated the facets corresponding to the rays, 
and dividing the surface in the usual symmetrical way. 


‘Fig. 234. The same, diagrammatic. 


31— 


484 ON CONCRETIONS, SPICULES, ETC. [cH. 


Within any polyhedron we may always inscribe another 
polyhedron, whose corners correspond in number to the sides or 
facets of the original figure, or (in alternative cases) to a certain 
number of these sides; and a similar result is obtained by bevelling 
_ off the corners of the original polyhedron. We may obtain a 
precisely similar symmetrical result if (in such a case as these 
Radiolarians which we are describing), we imagine the radial 
spines to be interconnected by tangential rods, instead of by the 
complete facets which we have just been dealing with. In our 
complicated polyhedron with its twenty radial spines arranged in 
the manner described there are various symmetrical ways in which 
we may imagine these interconnecting bars to be arranged. The 
most symmetrical of these is one in which the whole surface is 
divided into eighteen rhomboidal areas, obtained by systematically 
connecting each group of four adjacent radi. This figure has 
eighteen faces (F), twenty corners (C), and therefore thirty-six 
edges (#), in conformity with Euler’s theorem, F + C = E+ 2. 
Another symmetrical arrange- 
ment will divide the surface 
into fourteen rhombs and eight 
triangles. This latter arrange- 
ment is obtained by linking up 
the radial rods as follows: aaaa, 
aba, abcb, bedc, etc. Here we 
have again twenty corners, but 
we have twenty-two faces; the 
number of edges, or tangential 
spicular bars, will be found, 
therefore, by the above formula, 
to be forty. In Haeckel’s figure 
of Phractaspis prototypus we 
have a spicular skeleton which 
appears to be constructed precisely upon this plan, and to 
be derivable from the faceted polyhedron precisely after this 
manner. 

In all these latter cases it is the arrangement of the axial 
rods, or in other words the “polar symmetry” of the entire 
organism, which lies at the root of the matter, and which, if only 


Fig. 235. Phractaspis prototypus, Hk, 


1X] OF LIQUID OR FLUID CRYSTALS 485 


we could account for it, would make it comparatively easy to 
explain the superficial configuration. But there are no obvious 
mechanical forces by which we can so explain this peculiar 
polarity. This at least is evident, that it arises in the central 
mass of protoplasm, which is the essential living portion of the 
organism as distinguished from that frothy peripheral mass whose 
structure has helped us to explain so many phenomena of the 
superficial or external skeleton. To say that the arrangement 
depends upon a specific polarisation of the cell is merely to refer 
the problem to other terms, and to set it aside for future solution. 
But it is possible that we may learn something about the lines in 
which te seek for such a solution by considering the case of 
Lehmann’s “ fluid crystals.” and the light which they throw upon 
the phenomena of molecular aggregation. 

The phenomenon of “fluid crystallisation” is found in a 
number of chemical bodies; it is exhibited at a specific temperature 
for each substance; and it would seem to be limited to bodies 
in which there is a more or less elongated, or “chain-like” arrange- 
ment of the atoms in the molecule. Such bodies, at the appropriate 
temperature, tend to aggregate themselves into masses, which are 
sometimes spherical drops or globules (the so-called “spherulites”’), 
and sometimes have the definite form of needle-like or prismatic 
crystals. In either case they remain liquid, and are also doubly 

refractive, polarising light in brilliant colours. Together with 
- them are formed ordinary solid crystals, also with characteristic 
polarisation, and into such solid crystals all the fluid material 
ultimately turns. It is evident that in these liquid crystals, 
though the molecules are freely mobile, just as are those of water, 
they are yet subject to, or endowed with, a “directive force,” 
a force which confers upon them a definite configuration or 
“polarity,” the Gestaltungskraft of Lehmann. 

Such an hypothesis as this had been gradually extruded from 
the theories of mathematical crystallography*; and it had come 
to be believed that the symmetrical conformation of a homo- 
geneous crystalline structure was sufficiently explained by the 
mere mechanical fitting together of appropriate structural units 
along the easiest and simplest lines of “close packing”: just as 
* Cf. Tutton, Crystallography, p. 932, 1911. 


486 ON CONCRETIONS, SPICULES, [CH. 


a pile of oranges becomes definite, both in outward form and 
inward structural arrangement, without the play of any specific 
directive force. But while our conceptions of the tactical arrange- 
ment of crystalline molecules remain the same as before, and our 
hypotheses of “modes of packing” or of “space-lattices” remain 
as useful as ever for the definition and explanation of the 
molecular arrangements, an entirely new theoretical conception 
is introduced when we find such space-lattices maintained in 
what has hitherto been considered the molecular freedom of a 
liquid field; and we are constrained, accordingly, to postulate 
a specific molecular force, or “Gestaltungskraft”’ (not unlike 
Kepler’s “ facultas formatrix ’’), to account for the phenomenon. 
Now just as some sort of specific “Gestaltungskraft” had 
been of old the deus ex machina accounting for all crystalline 
phenomena (gnara totius geometrie, et in ea exercita, as Kepler 
said), and as such an hypothesis, after being dethroned and 
repudiated, has now fought its way back and has made good its 
right to be heard, so it may be also in biology. We begin by an 
easy and general assumption of specific properties, by which each 
organism assumes its own specific form; we learn later (as it is 
the purpose of this book to shew) that throughout the whole 
range of organic morphology there are innumerable phenomena of 
form which are not peculiar to living things, but which are more 
or less simple manifestations of ordinary physical law. But every 
now and then we come to certain deep-seated signs of proto-— 
plasmic symmetry or polarisation, which seem to lie beyond the 
reach of the ordinary physical forces. It by no means follows 
that the forces in question are not essentially physical forces, more 
obscure and less familiar to us than the rest; and this would seem 
to be the crucial lesson for us to draw from Lehmann’s surprising 
and most beautiful discovery. For Lehmann seems actually to 
have demonstrated, in non-living, chemical bodies, the existence 
of just such a determinant, just such a “Gestaltungskraft,” as 
would be of infinite help to us if we might postulate it for the 
explanation (for instance) of our Radiolarian’s axial symmetry. 
But further than this we cannot go; for such analogy as we seem 
to see in the Lehmann phenomenon soon evades us, and refuses 
to be pressed home. Not only is it the case, as we have already 


Ix] AND SPICULAR SKELETONS 487 


seen, that certain of the geometric forms assumed by the symme- 
trical Radiolarian shells are just such as the “space-lattice” 
theory would seem to be inapplicable to, but it is in other ways 
obvious that symmetry of crystallisation, whether liquid or solid, 
has no close parallel, but only a series of analogies, in the proto- 
plasmic symmetry of the living cell. 


CHAPTER X 
A PARENTHETIC NOTE ON GEODETICS 


We have made use in the last chapter of the mathematical 
principle of Geodetics (or Geodesics) in order to explain the con- 
formation of a certain class of sponge-spicules; but the principle 
is of much wider application in morphology, and would seem to 
deserve attention which it has not yet received. 

Defining, meanwhile, our geodetic line (as we have already 
done) as the shortest distance between two points on the surface 
of a solid of revolution, we find that the geodetics of the cylinder 
give us one of the simplest of 
cases. Here it is plain that the 
geodetics are of three kinds: (1) 
a series of annuli around the 
cylinder, that is to say, a system 
of circles, in planes parallel to 
one another and at right angles 
to the axis of the cylinder (Fig. 
236, a); (2) a series of straight 
lines parallel to the axis; and 
(3) a series of spiral curves wind- 
ing round the wall of the cylinder 

(b, c). These three systems are 
: B S ll of frequent occurrence, and 
Fig. 236. Annular and spiral thick- : 4 : ; 
enings in the walls of plant-cells. are all illustrated in the local 
thickenings of the wall of the 
cylindrical cells or vessels of plants. 

The spiral, or rather helicoid, geodetic is particularly common 
in cylindrical structures, and is beautifully shewn for instance in 
the spiral coil which stiffens the tracheal tubes of an insect, or 
the so-called “tracheides” of a woody stem. A similar pheno- 


2 


CH. x] ON GEODETICS 489 


menon is often witnessed in the splitting of a glass tube. If a 
crack appear in a thin tube, such as a test-tube, it has a tendency 
to be prolonged in its own direction, and the more perfectly 
homogeneous and isotropic be the glass the more evenly will the 
split tend to follow the straight course in which it began. As 
a result, the crack in our test-tube is often seen to continue till 
the tube is split into a continuous spiral ribbon. 

In a right cone, the spiral geodetic falls into closer and closer 
coils as the diameter of the cone narrows; and a very beautiful 
geodetic of this kind is exemplified in the sutural line of a spiral 
shell, such as Turritella, or in the striations which run parallel 
with the spiral suture. Similarly, in an ellipsoidal surface, we 
have a spiral geodetic, whose coils get closer together as we 
approach the ends of the long axis of the ellipse; in the splitting 
of the integument of. an Equisetum-spore, by which are formed 
the spiral “elaters” of the spore, we have a case of this kind, 
though the spiral is not sufficiently prolonged to shew all its 
features in detail. 

We have seen in these various cases, that our original definition 
of a geodetic requires to be modified; for it is only subject to 
conditions that it is “the shortest distance between two points 
on the surface of the solid,” and one of the commonest of these 
restricting conditions is that our geodetic may be constrained to 
go twice, or many times, round the surface on its way. In short, 
we must redefine our geodetic, as a curve drawn upon a surface, 
such that, if we take any two adjacent points on the curve, 
the curve gives the shortest distance between them. Again, 
in the geodetic systems which we meet with in morphology, it 
sometimes happens that we have two opposite systems of geodetic 
spirals separate and distinct from one another, as in Fig. 236, c; 
and it is also common to find the two systems interfering with 
one another, and forming a criss-cross, or reticulated arrangement. 
This is a very common source of reticulated patterns. 

Among the ciliated Infusoria, we have in the spiral lines along 
which their cilia are arranged a great variety of beautiful geodetic 
curves; though it is probable enough that in some complicated 
cases these are not simple geodetics, but projections of curves 
other than a straight line upon the surface of the solid. 


490 A PARENTHETIC NOTE [CH. 


Lastly, a very instructive case is furnished by the arrangement 
of the muscular fibres on the surface of a hollow organ, such as 
the heart or the stomach. Here we may consider the phenomenon 
from the point of view of mechanical efficiency, as well as from 
that of purely descriptive or objective anatomy. In fact we have 
an a priord right to expect that the muscular fibres covering such 
hollow or tubular organs will coincide with geodetic lines, in the 
sense in which we are now using the term. For if we imagine a 
contractile fibre, or elastic band, to be fixed by its two ends upon 
a curved surface, it is obvious that its first effort of contraction 
will tend to expend itself in accommodating the band to the 
form of the surface, in “stretching it tight,” or in other words 
in causing it to assume a direction which is the shortest possible 
line wpon the surface between the two extremes: and it is only 
then that further contraction will have the effect of constricting 
the tube and so exercising pressure on its contents. Thus the 
muscular fibres, as they wind over the curved surface of an organ, 
arrange themselves automatically in geodesic curves: in precisely 
the same manner as we also automatically construct complex 
systems of geodesics whenever we wind a ball of wool or a spindle 
of tow, or when the skilful surgeon bandages a limb. In these 
latter cases we see the production of those “figures-of-eight,” to 
which, in the case for instance of the heart-muscles, Pettigrew 
and other anatomists have ascribed peculiar importance. In the 
case of both heart and stomach we must look upon these organs 
as developed from a simple cylindrical tube, after the fashion of 
the glass-blower, as is further discussed on p. 737 of this book, 
the modification of the simple cylinder consisting of various degrees 
of dilatation and of twisting. In the primitive undistorted 
cylinder, as in an artery or in the intestine, the muscular fibres 
run in geodetic lines, which as a rule are not spiral, but are merely 
either annular or longitudinal; these are the ordinary “circular 
and longitudinal coats,’ which form the normal musculature of 
all tubular organs, or of the body-wall of a cylindrical worm*. If 
we consider each muscular fibre as an elastic strand, imbedded in 
the elastic membrane which constitutes the wall of the organ, it 


* However, we can often recognise, in a small artery for instance, that the so- 
called “circular” fibres tend to take a slightly oblique, or spiral, course. 


x] ON GEODETICS 491 


is evident that, whatever be the distortion suffered by the entire 
organ, the individual fibre will follow the same course, which will 
still, in a sense, be a geodetic. But if the distortion be consider- 
able, as for instance if the tube become bent upon itself, or if at 
some point its walls bulge outwards in a diverticulum or pouch, 
it is obvious that the old system of geodetics will only mark the 
shortest distance between two points more or less approximate to 
one another, and that new systems of geodetics will tend to 
appear, peculiar to the new surface, and linking up points more 
remote from one another. This is evidently the case in the 
human stomach. We still have the systems, or their unobliterated 
remains, of circular and longitudinal muscles; but we also see 
two new systems of fibres, both obviously geodetic (or rather. 
when we look more closely, both parts of one and the same 
geodetic system), in the form of annuli encircling the pouch or 
diverticulum at the cardiac end of the stomach, and: of oblique 
fibres taking a spiral course from the neighbourhood of the 
oesophagus over the sides of the organ. 


In the heart we have a similar, but more complicated 
phenomenon. Its musculature consists, in great part, of the 
original simple system of circular and longitudinal muscles 
which enveloped the original arterial tubes, which tubes, after 
a process of local thickening, expansion, and especially twisting, 
came together to constitute the composite, or double, mammalian 
heart; and these systems of muscular fibres, geodetic to begin 
with, remain geodetic (in the sense in which we are using the 
word) after all the twisting to which the primitive cylindrical tube 
or tubes have been subjected. That is to say, these fibres still 
run their shortest possible course, from start to finish, over the 
complicated curved surface of the organ; and it is only because 
they do so that their contraction, or longitudinal shortening, is 
able to produce its direct effect, as Borelli well understood, in 
the contraction or systole of the heart*. 


* The spiral fibres, or a large portion of them, constitute what Searle called 
“the rope of the heart” (Todd’s Cyclopaedia, u, p. 621, 1836), The ‘ twisted 
sinews of the heart” were known to early anatomists, and have been frequently 
and elaborately studied: for instance, by Gerdy (Bull. Fac. Med. Paris, 1820, 


492 A PARENTHETIC NOTE ON GEODETICS [cu. x 


“As a parenthetic corollary to the case of the spiral pattern 
upon the wall of a cylindrical cell, we may consider for a 
moment the spiral line which many small organisms tend to 
follow in their path of locomotion*. The helicoid spiral, traced 
around the wall of our cylinder, may be explained as a composition 
of two velocities, one a uniform velocity in the direction of the 
axis of the cylinder, the other a uniform velocity in a circle 
perpendicular to the axis. In a somewhat analogous fashion, the 
smaller ciated organisms, such as the ciliate and flagellate 
Infusoria, the Rotifers, the swarm-spores of various Protists, and 
so forth, have a tendency to combine a direct with a revolving 
path in their ordinary locomotion. The means of locomotion 
which they possess in their cilia are at best somewhat primitive 
and inefficient; they have no apparent means of steering, or 
modifying their direction; and, if their course tended to swerve 
ever so little to one side, the result would be to bring them round 
and round again in an approximately circular path (such as a man 
astray on the prairie is said to follow), with little or no progress 
in a definite longitudinal direction. But as a matter of fact, 
either through the direct action of their cilia or by reason of a 
more or less unsymmetrical form of the body, all these creatures 
tend more or less to rotate about their long axis while they swim. 
And this axial rotation, just as in the case of a rifle-bullet, causes 
their natural swerve, which is always in the same direction as 
regards their own bodies, to be in a continually changing direction 
as regards space: in short, to make a spiral course around, and 
more or less approximate to, a straight axial line. 


pp. 40-148), and by Pettigrew (Phil. Trans. 1864), and of late by J. B. Macallum 
(Johns Hopkins Hospital Report, 1x, 1900) and by Franklin P. Mall (Amer. J. of 
Anat. x1, 1911). 

* Cf. Biitschli, ‘‘ Protozoa,” in Bronn’s Thierreich, 11, p. 848, 11, p. 1785, etc., 
1883-87; Jennings, Amer. Nat. xxxv, p. 369, 1901; Piitter, Thigmotaxie bei 
Protisten, Arch. f. Anat. u. Phys. (Phys. Abth. Suppl.), pp. 243-302, 1900. 


LIBR AR Y¥ 


Quy) Ne » 7 


CHAPTER XI i A A 


THE LOGARITHMIC SPIRAL 


The very numerous examples of spiral conformation which we 
meet with in our studies of organic form are peculiarly adapted 
to mathematical methods of investigation. But ere we begin to 
study them, we must take care to define our terms, and we had 
better also attempt some rough preliminary classification of the 
objects with which we shall have to deal. 

In general terms, a Spiral Curve is a line which, starting from 
a point of origin, continually diminishes in curvature as it recedes 
from that point; or, in other words, whose radius of curvature 
continually increases. This definition is wide enough to include 
a number of different curves, but on the other hand it excludes 
at least one which in popular speech we are apt to confuse with 
a true spiral. This latter curve is the simple Screw, or cylindrical 
Helix, which curve, as is very evident, neither starts from a definite 
origin, nor varies in its curvature as it proceeds. The “spiral” 
thickening of a woody plant-cell, the “spiral” thread within an 
insect’s tracheal tube, or the “spiral” twist and twine of a climbing 
stem are not, mathematically speaking, spirals at all, but screws 
or helaces. They belong to a distinct, though by no means very 
remote, family of curves. Some of these helical forms we have 
just now treated of, briefly and parenthetically, under the subject 
of Geodetics. 

Of true organic spirals we have no lack*. We think at once 
of the beautiful spiral curves of the horns of ruminants, and of 
the still more varied, if not more beautiful, spirals of molluscan 
shells. Closely related spirals may be traced in the arrangement 

* A great number of spiral forms, both organic and artificial, are described 


and beautifully illustrated in Sir T. A. Cook’s Curves of Life, 1914, and Spirals in 
Nature and Art, 1903. : : 


Aga 97 THE LOGARITHMIC SPIRAL [on. 


of the florets in the sunflower; a true spiral, though not, by the 
way, so easy of investigation, is presented to us by the outline 


Fig. 237. The shell of Nautilus pompilius, from a radiograph: to shew the 
logarithmic spiral of the shell, together with the arrangement of the internal 
septa. (From Messrs Green and Gardiner, in Proc. Malacol. Soc. 1, 1897.) 


of a cordate leaf; and yet again, we can recognise typical though 
transitory spirals in the coil of an elephant’s trunk, in the “circling 


xt}! IN VARIOUS ORGANISMS 495 


spires’ of a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s 
or a chameleon’s tail. 

Among such forms as these, and the many others which we 
might easily add to them, it is obvious that we have to do with 
things which, though mathematically similar, are biologically 
speaking fundamentally different. And not only are they bio- 
logically remote, but they are also physically different, in regard 
to the nature of the forces to which they are severally due. For 
in the first place, the spiral coil of the elephant’s trunk or of the 
chameleon’s tail is, as we have said, but a transitory configuration, 
and is plainly the result of certain muscular forces acting upon 
a structure of a definite, and normally an essentially different, 
’ form. It is rather a position, or an attetude, than a form, in the 


Fig. 238. A Foraminiferal shell (Globigerina). 


sense in which we have been using this latter term; and, unlike 
most of the forms which we have been studying, it has little or no 
direct relation to the phenomenon of Growth. 

Again, there is a manifest and not unimportant difference 
between such a spiral conformation as is built up by the separate 
and successive florets in the sunflower, and that which, in the 
snail or Nautilus shell, is apparently a single and indivisible unit. 
And a similar, if not identical difference is apparent between the 
Nautilus shell and the minute shells of the Foraminifera, which 
so closely simulate it; inasmuch as the spiral shells of these latter 
are essentially composite structures, combined out of successive 
and separate chambers, while the molluscan shell, though it may 
(as in Nautilus) become secondarily subdivided, has grown as 
one continuous tube. It follows from all this that there cannot 


496 THE LOGARITHMIC SPIRAL . [CH. 


possibly be a-physical or dynamical, though there may well be 
a mathematical Law of Growth, which is common to, and which 
defines, the spiral form in the Nautilus, in the Globigerina, in the 
ram’s horn, and in the dise of the sunflower. 

Of the spiral forms which we have now mentioned, every one 
(with the single exception of the outline of the cordate leaf) is an 
example of the remarkable curve known as the Logarithmic Spiral. 
But before we enter upon the mathematics of the logarithmic 
spiral, let us carefully observe that the whole of the organic forms 
in which it is clearly and permanently exhibited, however different 
they may be from one another in outward appearance, in nature 
and in origin, nevertheless all belong, in a certain sense, to one 
particular class of conformations. In the great majority of cases, 
when we consider an organism in part or whole, when we look (for 
instance) at our own hand or foot, or contemplate an insect or 
a worm, we have no reason (or very little) to consider one part 
of the existing structure as older than another; through and 
through, the newer particles have been merged and commingled, 
by intussusception, among the old; the whole outline, such as it 
is, is due to forces which for the most part are still at work to 
shape it, aftd which in shaping it have shaped it as a whole. But 
the horn, or the snail-shell, is curiously different; for in each of 
these, the presently existing structure is, so to speak, partly old 
and partly new; it has been conformed by successive and con- 
tinuous increments; and each successive stage of growth, starting 
from the origin, remains as an integral and unchanging portion 
of the still growing structure, and so continues to represent what 
at some earlier epoch constituted for the time being the structure 
in its entirety. 

In a slightly different, but closely cognate way, the same is 
true of the spirally arranged florets of the sunflower. For here 
again we are regarding serially arranged portions of a composite 
structure, which portions, similar to one another in form, differ 
nage; and they differ also in magnitude in a strict ratio according 
to their age. Somehow or other, in the logarithmic spiral the 
time-element always enters in; and to this important fact, full of 
curious biological as well as mathematical significance, we shall 
afterwards return. 


x1] ITS GENERAL PROPERTIES 497 


It is, as we have so often seen, an essential part of our whole 
problem, to try to understand what distribution of forces is capable 
of producing this or that organic form,—to give, in short, a 
dynamical expression to our descriptive morphology. Now the 
general distribution of forces which lead to the formation of a 
spiral (whether logarithmic or other) is very easily understood ; 
and need not carry us beyond the use of very elementary mathe- 
matics. 

If we imagine growth to actin a perpendicular direction, as for 
example the upward force of growth in a growing stem (OA), then, 


Fig. 239. 


in the absence of other forces, elongation will as a matter of course 
proceed in an unchanging direction, that is to say the stem will 
grow straight upwards. Suppose now that there be some constant 
external force, such as the wind, impinging on the growing stem; 
and suppose (for simplicity’s sake) that this external force be in « 
constant direction (AB) perpendicular tothe intrinsic force of growth. 
The direction of actual growth will be in the line of the resultant 
of the two forces: and, since the external force is (by hypothesis) 
constant in direction, while the internal force tends always to act in 
the line of actual growth, it is obvious that our growing organism 
will tend to be bent into a curve, to which, for the time being, 


T. G. 32 


498 THE LOGARITHMIC SPIRAL | [CH. 


the actual force of growth will be acting at a tangent. So long 
as the two forces continue to act, the curve will approach, but 
will never attain, the direction of AB, perpendicular to the original 
direction OA. If the external force be constant in amount the 
curve will approximate to the form of a hyperbola; and, at any 
rate, it is obvious that it will never tend to assume a spiral 
form. 

In like manner, if we consider a horizontal beam, fixed at one 
end, the imposition of a weight at the other will bend the beam 
into a curve, which, as the beam elongates or the weight increases, 
will bring the weighted end nearer and nearer to the vertical. 
But such a force, constant in direction, will obviously never curve 
the beam into a spiral,—a fact so patent and obvious that it would 
be superfluous to state it, were it not that some naturalists have 
been in the habit of invoking gravity as the force to which ae be 
attributed the spiral flexure of the shell. 

But if, on the other hand, the deflecting force be cnherent in 
the growing body, or so connected with it in a system that its 
direction (instead of being constant, as in the former case) changes 
with the direction of growth, and is perpendicular (or inclined at 
some constant angle) to this changing direction of the growing 
force, then it is plain that there is no such limit to the deflection 
from the normal, but the growing curve will tend to wind round 
and round its point of origin. In the typical case of the snail- 
shell, such an intrinsic force is manifestly present in the action 
of the columellar muscle. 

Many other simple illustrations can be given of a spiral course 
being impressed upon what is primarily rectilinear motion, by 
any steady deflecting force which the moving body carries, so 
to speak, along with it, and which continually gives a lop-sided 
tendency to its forward movement. For instance, we have been told 
that a man or a horse, travelling over a great prairie, is very apt 
to find himself, after a long day’s journey, back again near to his 
starting point. Here some small and imperceptible bias, such as 
might for instance be caused by one leg being in a minute degree 
longer or stronger than the other, has steadily deflected the forward 
movement to one side; and has gradually brought the traveller 
back, perhaps in a circle to the very point from which he set out, 


xt] ITS GENERAL PROPERTIES 499 


or else by a spiral curve, somewhere within reach and recognition 
of it. 

We come to a similar result when we consider, for instance, 
a cylindrical body in which forces of growth are at work tending 
to its elongation, but these forces are unsymmetrically distributed. 
Let the tendency to elongation along AB be of a magnitude pro- 
portional to BB’, and that along CD be of a magnitude proportional 
to DD’; and in each element parallel to AB and CD, let a parallel 
force of growth, proportionately intermediate in magnitude, be at 
work: and let EFF’ be the middle line. Then at any cross- 
section BFD, if we deduct the mean force FF’, we have a certain 
positive force at B, equal to Bb, and an equal and opposite force 
at D, equal to Dd. But AB and CD are not separate structures, 


Ud ’ 
B F , B FE - 
‘ 
b D’ 
B E B E D 
d 
G 
b 
d 
Fig. 240. 


but are connected together, either by a solid core, or by the walls 
of a tubular shell; and the forces which tend to separate B and 
D are opposed, accordingly, by a tension in BD. It follows there- 
fore, that there will be a resultant force BG, acting in a direction 
intermediate between Bb and BD, and also a resultant, DH, 
acting at D in an opposite direction; and accordingly, after a 
small increment of growth, the growing end of the cylinder will 
come to lie, not in the direction BD, but in the direction GH, 
The problem is therefore analogous to that of a beam to which 
we apply a bending moment; and it is plain that the unequal 
force of growth is equivalent to a “couple” which will impart to 
our structure a curved form. For, if we regard the part ABDC 
as practically rigid, and the part BB’D’D as pliable, this couple 


32—2 


500 THE LOGARITHMIC SPIRAL [CH. 


will tend to turn strips such as B’D’ about an axis perpendicular 
to the plane of the diagram, and passing through an intermediate 
point F’. It is plain, also, since all the forces under consideration 
are intrinsic to the system, that this tendency will be continuous, 
and that as growth proceeds the curving body will assume either 
a circular or a spiral form. But the tension which we have here 
assumed to exist in the direction BD will obviously disappear if 
we suppose a sufficiently rapid rate of growth in that direction. 
For if we may regard the mouth of our tubular shell as perfectly 
extensible in its own plane, so that it exerts no traction whatsoever 
on the sides, then it will be drawn out into more and more elongated 
ellipses, forming the more and more oblique orifices of a straight 
tube. In other words, in such a structure as we have presupposed, 


Fig. 241. 


the existence or maintenance of a constant ratio between the 
rates of extension or growth in the vertical and transverse directions 
will lead, in general, to the development of a logarithmic spiral; 
the magnitude of that ratio will determine the character (that is 
to say, the constant angle) of the spiral; and the spirals so pro- 
duced will include, as special or limiting cases, the circle and the 
straight line. 

We may dispense with the hypothesis of bending moments, 
if we simply presuppose that the increments of growth take place 
at a constant angle to the growing surface (as AB), but more 
rapidly at A (which we shall call the “outer edge”) than at B, 
and that this difference of velocity maintains a constant ratio. 
Let us also assume that the whole structure is ngid, the new 
accretions solidifying as soon as they are laid on. For example, 


xt] - ITS GENERAL PROPERTIES 501 


let Fig. 242 represent in section the early growth of a Nautilus- 
shell, and let the part ARB represent the earliest stage of all, 
which in Nautilus is nearly semicircular. We have to find a law 
governing the growth of the shell, such that each edge shall 
develope into an equiangular spiral; and this law, accordingly, 
must be the same for each edge, namely that at each instant the 
direction of growth makes a constant angle with a line drawn from 
a fixed point (called the pole of the spiral) to the point at which 
growth is taking place. This growth, we now find, may be 
considered as effected by the continuous addition of similar 
quadrilaterals. Thus, in Fig. 241, AHDB is a quadrilateral with 
AE, DB parallel, and with the angle HAB of a certain definite 


; - 
Fig. 242. 


magnitude, = y. Let AB and HD meet, when produced, in C; 
and call the angle ACE (or xCy) = 8. Make the angle yCz = angle 
xCy,= 8. Draw EG, so that the angle yEG = y, meeting Cz in 
G; and draw DF parallel to HG. It is then easy to show that 
AEDB and EGFD are similar quadrilaterals. And, when we 
consider the quadrilateral AHDB as having infinitesimal sides, 
AE and BD, the angle y tends to a, the constant angle of an equi- 
angular spiral which passes through the points AHG, and of a 
similar spiral which passes through the points BDF; and the point 
C is the pole of both of these spirals. In a particular limiting case, 
when our quadrilaterals are all equal as well as similar,—which 
will be the case when the angle y (or the angles HAC, etc.) is a 


502 THE LOGARITHMIC SPIRAL _ [on. 


right angle,—the “spiral” curve will be a circular arc, C being the 
centre of the circle. 


Another, and a very simple illustration may be drawn from the ‘‘cymose 
inflorescences” of the botanists, though the actual mode of development of 
some of these structures is open to dispute, and their nomenclature is involved 
in extraordinary historical confusion *. 

In Fig. 243 B (which represents the Cicinnus of Schimper, or cyme unipare 
scorpioide of Bravais, as seen in the Borage), we begin with a primary shoot 

from which is given off, at a certain definite 
angle, a secondary shoot: and from that in turn, 
on the same side and at the same angle, another 
shoot, and so on. The deflection, or curvature, 
is continuous and progressive, for it is caused by 
no external force but only by causes intrinsic in 
a) the system. And the whole system is sym- 
metrical: the angles at which the successive 
shoots are given off being all equal, and the 


BS lengths of the shoots diminishing in constant 

ratio. The result is that the successive shoots, 

A B or successive increments of growth, are tangents 

Hip, 243;. Ala belived,’ B to a curve, and this curve is a true logarith- 
ae scorpioid cyme. mic spiral. But while, in this simple case, 


the successive shoots are depicted as lying in 
a plane, it may also happen that, in addition to their successive angular 
divergence from one another within that plane, they also tend to diverge 
by successive equal angles from that plane of reference; and by this 
means, there will be superposed upon the logarithmic spiral a helicoid twist 
or screw. And, in the particular case where this latter angle of divergence 
is just equal to 180°, or two right angles, the successive shoots will once more 
come to lie in a plane, but they will appear to come off from one another on 
alternate sides, as in Fig. 243 A” This is the Schraubel or Bostryx of Schimper, 
the cyme unipare hélicoide of Bravais. The logarithmic spiral is still latent 
in it, as in the other; but is concealed from view by the deformation resulting 
from the helicoid. The confusion of nomenclature would seem to have arisen 
from the fact that many botanists did not recognise (as the brothers Bravais did) 
the mathematical significance of the latter case; but were led, by the snail- 
like spiral of the scorpioid cyme, to transfer the name “‘helicoid” to it. 


In the study of such curves as these, then, we speak of the 
point of origin as the pole (O); a straight line having its extremity 
in the pole and revolving about it, is called the radius vector ; 


* Cf. Vines, The History of the Scorpioid Cyme, Journ. of Botany (n.s.), X, 
pp. 3-9, 1881. = 


x1] AND THE SPIRAL OF ARCHIMEDES 503 


and a point (P) which is conceived as travelling along the radius 
vector under definite conditions of velocity, will then describe our 
spiral curve. 

Of several mathematical curves whose form and development 
may be so conceived, the two most important (and the only two 
with which we need deal), are those which are known as (1) the 
equable spiral, or spiral of Archimedes, and (2) the logarithmic, 
or equiangular spiral. 

The former may be illustrated by the spiral coil in which a 
sailor coils a rope upon the deck; as the rope is of uniform thick- 
ness, so in the whole spiral coil is each whorl of the same breadth 


Fig. 244 


as that which precedes and as that which follows it. Using 
its ancient definition, we may define it by saying, that “If a 
straight line revolve uniformly about its extremity, a point which 
hkewise travels uniformly along it will describe the equable 
spiral*.”” Or, putting the same thing into our more modern 
words, “If, while the radius vector revolve uniformly about the 
pole, a point (P) travel with uniform velocity along it, the curve 
described will be that called the equable spiral, or spiral of 
Archimedes.” 

* Leslie’s Geometry of Curved Lines, p. 417, 1821. This is practically identical 


with Archimedes’ own definition (ed. Torelli, p. 219); ef. Cantor, Geschichte der 
Mathematik, 1, p. 262, 1880. 


504 THE LOGARITHMIC SPIRAL ee 3 


It is plain that the spiral of Archimedes may be compared to 
a cylinder coiled up. And it is plain also that a radius (r = OP), 
made up of the successive and equal whorls, will increase in 
arithmetical progression: and will equal a certain constant 
quantity (a) multiplied by the whole number of whorls, or (more 
strictly speaking) multiplied by the whole angle (#@) through 
which it has revolved: so that r= a0. 

But, in contrast to this, in the logarithmic spiral of the Nau- 
tilus or the snail-shell, the whorls gradually increase in breadth, 
and do so in a steady and unchanging ratio. Our definition is 
as follows: “If, instead of travelling with-a uniform velocity, 
our point move along the radius vector with a velocity increasing 
as its distance from the pole, then the path described is called a 
logarithmic spiral.” Hach whorl which the radius vector inter- 
sects will be broader than its predecessor in a definite ratio; the 
radius vector will increase in length in geometrical progression, 
as it sweeps through successive equal angles; and the equation 
to the spiral will be r =a’. As the spiral of Archimedes, in our 
example of the coiled rope, might be looked upon as a coiled 
cylinder, so may the logatrithmic spiral, in the case of the shell, 
be pictured as a cone coiled upon itself. 

Now it is obvious that if the whorls increase very slowly indeed, 
the logarithmic spiral will come to look like a spiral of Archimedes, 
with which however it never becomes identical; for it is incorrect 
to say, as is sometimes done, that the Archimedean spiral is a 
“limiting case” of the logarithmic spiral. The Nummulite is a 
case in point. Here we have a large number of whorls, very 
narrow, very close together, and apparently of equal breadth, 
which give rise to an appearance similar to that of our coiled 
rope. And, in a case of this kind, we might actually find that 
the whorls were of equal breadth, being produced (as is apparently 
the case in the Nummulite) not by any very slow and gradual 
growth in thickness of a continuous tube, but by a succession of 
similar cells or chambers laid on, round and round, determined as 
to their size by constant surface-tension conditions and there- 
fore of unvarying dimensions. But even in this case we should 
have no Archimedean spiral, but only a logarithmic spiral in 
which the constant angle approximated to 90°. 


XI] IN ITS DYNAMICAL ASPECT 505 


For, in the logarithmic spiral, when a tends to 90°, the expression 7 = qi cote 


tends to r= a(1+ @cot a); while the equation to the Archimedean spiral is 
r=b@. The nummulite must always have a central core, or initial cell, 
around which the coil is not only wrapped, but out of which it springs; and 
this initial chamber corresponds to our q@’ in the expression 7 = a’ + a cot a. 
The outer whorls resemble those of an Archimedean spiral, because of the 
other term aé cot a in the same expression. It follows from this that in all 
such cases the whorls must be of excessively small breadth. 


There are many other specific properties of the logarithmic 
spiral, so interrelated to one another that we may choose pretty 
well any one of them as the basis of our definition, and deduce the 
others from it either by analytical methods or by the methods of 
elementary geometry. For instance, the equation r = a’ may be 
written in the form log r = @ log a, or 6 = log r/log a, or (since @ is 
a constant), 0=klogr. Which is as much as to say that the 
vector angles about the pole are proportional to the logarithms 
of the successive radii; from which circumstance the name of the 
“logarithmic spiral” is derived. 


Let us next regard our logarithmic spiral from the dynamical 
point of view, as when we consider the forces concerned in the 
growth of a material, concrete spiral. 
Tn a growing structure, let the forces of 
growth exerted at any point P be a 
force F acting along the line joining P 
to a pole O and a force T acting in a 
direction perpendicular to OP; and let 
the magnitude of these forces be in the 
same constant ratio at all points. It 
follows that the resultant of the forces 
F and T (as PQ) makes a constant 
angle with the radius vector. But the Roe 
constancy of the angle between tangent Fig. 245. 
and radius vector at any point is a 
fundamental property of the logarithmic spiral, and may be 
shewn to follow from our definition of the curve: it gives to the 
curve its alternative name of equiangular spiral. Hence in a 
structure growing under the above conditions the form of the 
boundary will be a logarithmic spiral. 


506 THE LOGARITHMIC SPIRAL [CH. 


In such a spiral, radial growth and growth in the direction of 
the curve bear a constant ratio to one another. For, if we consider 
a consecutive radius vector, OP’, whose increment 
as compared with OP is dr, while ds is the small 
arc PP’, then 


dr/ds = cos a = constant. 


In the concrete case of the shell, the distribution 
of forces will be, originally, a little more compli- 
cated than this, though by resolving the forces in 
question, the system may be reduced to this 
simple form. And furthermore, the actual distri- 
bution of forces will not always be identical; 
for example, there is a distinct difference between the cases (as 
in the snail) where a columellar muscle exerts a definite traction 
in the direction of the pole, and those (such as Nautilus) where 
there is no columellar muscle, and where some other force must 
be discovered, or postulated, to account for the flexure. In the 

B most frequent case, we have, as 
aye in Fig. 247, three forces to deal 
| with, acting at a point, p: 
L, acting in the direction of 
the tangent to the curve, and - 
representing the force of longi- 
tudinal growth; TZ, perpen- 
dicular to L, and representing 
the organism’s tendency to grow 
in breadth; and P, the traction 
exercised, in the direction of the 
pole, by the columellar muscle. 
Let us resolve Z and T into 
components along P (namely 
A’, B’), and perpendicular to P (namely A, B); we have now only 
two forces to consider, viz. P— A’ — B’, and A— B. And these 
two latter we can again resolve, if we please, so as to deal only 
with forces in the direction of P and T. Now, the ratio of these 
forces remaining constant, the locus of the point p is an equiangular 
spiral. 


Fig. 246. 


Oo 
Fig. 247. 


xt] ITS MATHEMATICAL PROPERTIES 5OT 


Furthermore we see how any slight change in any one of the 
forces P, T, L will tend to modify the angle a, and produce a slight 
departure from the absolute regularity of the logarithmic spiral. 
Such slight departures from the absolute simplicity and uniformity 
of the theoretic law we shall not be surprised to find, more or less 
frequently, in Nature, in the complex system of forces presented 
by the living organism. 

In the growth of a shell, we can conceive no simpler law than 
this, namely, that it shall widen and lengthen in the same unvarying 
proportions: and this simplest of laws is that which Nature tends 
to follow. The shell, like the creature within it, grows in size 
but does not change its shape; and the existence of this constant 
relativity of growth, or constant similarity of form, is of the essence, 
and may be made the basis of a definition, of the logarithmic 
spiral. 

Such a definition, though not commonly used by mathe- 
maticians, has been occasionally employed; and it is one from 
which the other properties of the curve can be deduced with 
great ease and simplicity. In mathematical language it would run 
as follows: “Any [plane] curve proceeding from a fixed point 
(which is called the pole), and such that the arc intercepted between 
this point and any other whatsoever on the curve is always similar 
to itself, is called an equiangular, or logarithmic, spiral*.” 

In this definition, we have what is probably the most funda- 
mental and “intrinsic” property of the curve, namely the property 
of continual similarity: and this is indeed the very property by 
reason of which it is peculiarly associated with organic growth in 
such structures as the horn or the shell, or the scorpioid cyme 
which is described on p. 502. For it is peculiarly characteristic 
of the spiral of a shell, for instance, that (under all normal circum- 
stances) it does not alter its shape as it grows; each increment 1s 
geometrically similar to its predecessor, and the whole, at any 
epoch, is similar to what constituted the whole at another and an 
earlier epoch. We feel no surprise when the animal which secretes 
the shell, or any other animal whatsoever, grows by such sym- 


* See an interesting paper by Whitworth, W. A., “The Equiangular Spiral, 
its chief properties proved geometrically,” in the Messenger of Mathematics (1), 
1, p. 5, 1862. 


508 THE LOGARITHMIC SPIRAL [CH. 


metrical expansion as to preserve its form unchanged; though 
even there, as we have already seen, the unchanging form denotes 
a nice balance between the rates of growth in various directions, 
which is but seldom accurately maintained for long. But the 
shell retains its unchanging form in spite of its asymmetrical 
growth; it grows at one end only, and so does the horn. And 
this remarkable property of increasing by terminal growth, but 
nevertheless retaining unchanged the form of the entire figure, is 
characteristic of the logarithmic spiral, and of no other mathe- 
matical curve. 

We may at once illustrate this curious phenomenon by drawing 
the outline of a little Nautilus shell within a big one. We know, 
or we may see at once, that they are of precisely the same shape; 
so that, if we look at the little shell through a magnifying glass, 
it becomes identical with the big one. But we know, on the other 


Fig. 248. 


hand, that the little Nautilus shell grows into the big one, not by 
uniform growth or magnification in all directions, as is (though 
only approximately) the case when the boy grows into the man, 
but by growing at one end only. 


Though of all curves, this property of continued similarity is 
found only in the logarithmic spiral, there are very many rectilinear 
figures in which it may be observed. For instance, as we may 
easily see, it holds good of any right cone; for evidently, in Fig. 248, 
the little inner cone (represented in its triangular section) may 
become identical with the larger one either by magnification all 
round (as in a), or simply by an increment at one end (as in 6); 
indeed, in the case of the cone, we have yet a third possibility, 
for the same result is attained when it increases all round, save 
only at the base, that is to say when the triangular section increases 


XI] CONCERNING GNOMONS 509 


on two of its sides, as inc. All this is closely associated with the 
fact, which we have already noted, that the Nautilus shell is but 
a cone rolled up; in other words, the cone is but a particular 
variety, or “limiting case,” of the spiral shell. 

This property, which we so easily recognise in the cone, would 
seem to have engaged the particular attention of the most ancient 
mathematicians even from the days of Pythagoras, and so, with 
little doubt, from the more ancient days of that Egyptian school 
whence he derived the foundations of his learning* ; and its bearing 
on our biological problem of the shell, though apparently indirect, 
is yet so close that it deserves our further consideration. 

If, as in Fig. 249, we add to two sides of a square a symmetrical 
L-shaped portion, similar in shape to what we call a “ carpenter’s 
square,” the resulting figure is still a square; and the portion 


Fig. 249. Fig. 250. 
which we have added is called, by Aristotle (Phys. m1, 4), a 
“gnomon.” Kuclid extends the term to include the case of any 
parallelogram}, whether rectangular or not (Fig. 250); and Hero 
of Alexandria specifically defines a “gnomon” (as indeed Aristotle 
implicitly defines it), as any figure which, being added to any 
figure whatsoever, leaves the resultant figure similar to the 
original. Included in this important definition is the case of 
numbers, considered geometrically; that is to say, the eiéntvxol 
apOuot, which can be translated into form, by means of rows of 
dots or other signs (cf. Arist. Metaph. 1092b12), or in the 
pattern of a tiled floor: all according to “the mystical way of 

* T am well aware that the debt of Greek science to Egypt and the East is 
vigorously denied by many scholars, some of whom go so far as to believe that the 
Egyptians never had any science, save only some “rough rules of thumb for measur- 


ing fields and pyramids” (Burnet’s Greek Philosophy, 1914, p. 5). 
7 Euclid (u, def. 2). 


510 THE LOGARITHMIC SPIRAL [CH. 


Pythagoras, and the secret magick of numbers.” Thus for 
example, the odd numbers are “gnomonic numbers,” because 


0+1=1 
12+ 3 = 22, 
2245 = 3, 


a7 + 7 = 47 etc., 


which relation we may illustrate graphically (cynwatoypadetv) 
by the successive numbers of dots which keep the annexed figure 


a perfect square*: as follows: l 


There are other gnomonic figures more curious still. For 
instance, if we make a rectangle (Fig. 251) such that the two 


r/o r/o 
Vo 


1 
Fig. 251. ' 


sides are in the ratio of 1:V. 9, it is obvious that, on doubling it, 
we obtain a precisely similar figure; for 1:V2::V2:2; and 


* Cf. Treutlein, Z. f. Math. u. Phys. (Hist. litt. Abth.), xxvii, p. 209, 1883. 


x1| CONCERNING GNOMONS 511 


each half of the figure, accordingly, is now a gnomon to the other. 
Another elegant example is when we start with a rectangle (A) 
whose sides are in the proportion of 1: }(V 5 Lyon approxi- 
mately, 1: 0-618. The gnomon to this figure is a square (B) erected 
on its longer side, and so on successively (Fig. 252). 

In any triangle, as Aristotle tells us, one part is always a 
gnomon to the other part. For instance, in the triangle ABC 
(Fig. 253), let us draw CD, so as to make the angle BCD equal to 
the angle A. Then the part BCD is a triangle similar to the 
whole triangle ABC, and ADC is a gnomon to BCD. A very 
elegant case is when the original triangle ABC is an isosceles 
triangle having one angle of 36°, and the other two angles, there- 
fore, each equal to 72° (Fig. 254). Then, by bisecting one of the 


A A 


D 


ue 


B C B Cc 
Fig. 253. Fig. 254. 


angles of the base, we subdivide the large isosceles triangle into 
two isosceles triangles, of which one is similar to the whole figure 
and the other is its gnomon*. There is good reason to believe 
that this triangle was especially studied by the Pythagoreans ; 
for it les at the root of many interesting geometrical constructions, 
such as the regular pentagon, and the mystical “ pentalpha,” and 
a whole range of other curious figures beloved of the ancient 
mathematicians 7. 

* This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra 
cit. The ratio 1: 0-618 is again not hard to find in this construction. 

+ See, on the mathematical history of the Gnomon, Heath’s Euclid, 1, passim, 
1908; Zeuthen, Theoréme de Pythagore, Genéve, 1904; also a curious and 
interesting book, Das Theorem des Pythagoras, by Dr H. A. Naber, Haarlem, 1908. 


. 
4 


512 THE LOGARITHMIC SPIRAL [CH. 


If we take any one of these figures, for instance the isosceles 


A 


Fig. 255. 


triangle which we have just described, 
and add to it (or subtract from it) in 
succession a series of gnomons, so con- 
verting it into larger and larger (or smaller 
and smaller) triangles all similar to the 
first, we find that the apices (or other 
corresponding points) of all these triangles 
have their locus upon a logarithmic spiral : 
a result which follows directly from that 
alternative definition of the logarithmic 
spiral which I have quoted from Whit- 
worth (p. 507). 

Again, we may build up a series of 
right-angled triangles, each of which is a 


gnomon to the preceding figure; and here again, a logarithmic 
spiral is the locus of corresponding points in these successive 
triangles. And lastly, whensoever we fill up space with a 


a 
Lae: 


Bs SUN 


The. 256. Logarithmic spiral derived from corresponding points in 


a system of squares. 


xt] CONCERNING GNOMONS 513 


collection of either equal or similar figures, similarly situated, 
as in Figs. 256, 257, there we can always discover a series of 
inscribed or escribed logarithmic spirals. 

Once more, then, we may modify our definition, and say that: 
“ Any plane curve proceeding from a fixed point (or pole), and such 
that the vectorial area of any sector is always a gnomon to the 
whole preceding figure, is called an equiangular, or logarithmic, 
spiral.” And we may now introduce this new concept and 
nomenclature into our description of the Nautilus shell and 
other related organic forms, by saying that: (1) if a growing 


Fig. 257. The same in a system of hexagons. 


structure be built up of successive parts, similar and similarly 
situated, we can always trace through corresponding points 
a series of logarithmic spirals (Figs. 258, 259, ete.); (2) it is 
characteristic of the growth of the horn, of the shell, and of 
all other organic forms in which a logarithmic spiral can be 
recognised, that each successive increment of growth is a gnomon 
to the entire pre-existing structure. And conversely (3) it follows 
obviously, that in the logarithmic spiral outline of the shell 
or of the horn we can always inscribe an endless variety of 
other gnomonic figures, having no necessary relation, save as a 


T. Ge 33 


514 THE LOGARITHMIC SPIRAL [CH. 


mathematical accident, to the nature or mode of development 
of the actual structure*. 


Fig. 258. A shell of Haliotis. with two of the many lines of growth, or generating 
curves, marked out in black: the areas bounded by these lines of growth being 


in all cases ““gnomons”’ to the pre-existing shell. 


Fig. 259. A spiral foraminifer (Pulvinulina),.to show how each successive chamber 
continues the symmetry of, or constitutes a gnomon to, the rest of the structure. 


* For many beautiful geometrical constructions based on the molluscan shell, 
see Colman, S. and Coan, C. A., Nature’s Harmonic Unity (ch. ix, Conchology), 
New York, 1912. 


x1] CONCERNING GNOMONS 515 


Of these three propositions, the second is of very great use 
and advantage for our easy understanding and simple description 
of the molluscan shell, and of a great variety of other structures 
whose mode of growth is analogous, and whose mathematical 
properties are therefore identical. We see at once that the 
successive chambers of a spiral Nautilus (Fig. 237) or of a straight 
Orthoceras (Fig. 300), each whorl or part of a whorl of a peri- 
winkle or other gastropod (Fig. 258), each new increment of the 
operculum of a gastropod (Fig. 263), each additional increment of 
an elephant’s tusk, or each new 
chamber of a spiral foraminifer 
(Figs. 259 and 260), has its leading 
characteristic at once described and 
its form so far explained by the 
simple statement that it constitutes 
a gnomon to the whole previously 
existing structure. And herein lies 
the explanation of that “time- 
element” in the development of 
organic spirals of which we have 
spoken already, in a_ preliminary 
and empirical way. For it follows 
as a simple corollary to this 
theorem of gnomons that we must not expect to find the 
logarithmic spiral manifested in a structure whose parts are 
simultaneously produced, as for instance in the margin of a 
leaf, or among the many curves that make the contour of a 
fish. But we must rather look for it wherever the organism 
retains for us, and still presents to us at a single view, the successive 
phases of preceding growth, the successive magnitudes attained, 
the successive outlines occupied, as the organism or a part thereof 
pursued the even tenour of its growth, year by year and day by 
day. And it easily follows from this, that it is in the hard parts 
of organisms, and not the soft, fleshy, actively growing parts, 
that this spiral is commonly and characteristically found; not 
in the fresh mobile tissues whose form is constrained merely by 
the active forces of the moment; but in things like shell and tusk, 
and horn and claw, where the object is visibly composed of parts 

33—2 


Fig. 260. Another spiral fora- 
minifer, Cristellaria. 


516 THE LOGARITHMIC SPIRAL [CH. 


successively, and permanently, laid down. In the main, the 
logarithmic spiral is characteristic, not of the living tissues, but 
of the dead. And for the same reason, it will always or nearly 
always be accompanied, and adorned, by a pattern formed of 
“lines of growth,” the lasting record of earlier and successive 
stages of form and magnitude. 


It is evident that the spiral curve of the shell is, in a sense, 
a vector diagram of its own growth; for it shews at each instant 
of time, the direction, radial and tangential, of growth, and the 
unchanging ratio of velocities in these directions. Regarding the 
actual velocity of growth in the shell, we know very little (or 
practically nothing), by way of experimental measurement; but 
if we make a certain simple assumption, then we may go a good 
deal further in our description of the logarithmic spiral as it appears 
in this concrete case. 

Let us make the assumption that similar increments are added 
to the shell in equal times; that is to say, that the amount of 
growth in unit time is measured by the areas subtended by equal 
angles. Thus, in the outer whorl of a spiral shell a definite area 
marked out by ridges, tubercles, etc., has very different linear 
dimensions to the corresponding areas of the inner whorl, but the 
symmetry of the figure implies that it subtends an equal angle 
with these; and it is reasonable to suppose that the successive 
regions, marked out in this way by successive natural boundaries 
or patterns, are produced in equal intervals of time. 

If this be so, the radii measured from the pole to the boundary 
of the shell will in each case be proportional to the velocity of 
growth at this point upon the circumference, and at the time when 
it corresponded with the outer lip, or region of active growth; 
and while the direction of the radius vector corresponds with the 
direction of growth in thickness of the animal, so does the tangent 
to the curve correspond with the direction, for the time being, of 
the animal’s growth in length. The successive radii are a measure 
of the acceleration of growth, and the spiral curve of the shell 
itself is no other than the hodograph of the growth of the contained 
organism. 


x1] ITS MATHEMATICAL PROPERTIES 517 


So far as we have now gone, we have studied the elementary 
properties of the logarithmic spiral, including its fundamental 
property of continued similarity; and we have accordingly learned 
that the shell or the horn tends necessarily to assume the form 
of this mathematical figure, because in these structures growth 
proceeds by successive increments, which are always similar in 
form, similarly situated, and of constant relative magnitude one 
to another. Our chief objects in enquiring further into the 
mathematical properties of the logarithmic spiral will be: (1) to 
find means of confirming and verifying the fact that the shell (or 
other organic curve) is actually a logarithmic spiral; (2) to learn 
how, by the properties of the curve, we may further extend our 
knowledge or simplify our descriptions of the shell; and (3) to 
understand the factors by which the characteristic form of any 
particular logarithmic spiral is determined, and so to comprehend 
the nature of the specific or generic characters by which one spiral 
shell is found to differ from another. 

Of the elementary properties of the logarithmic spiral, so far as 
we have now enumerated them, the following are those which we 
may most easily investigate in the concrete case, such as we have 
to do with in the molluscan shell: (1) that the polar radii of points 
whose vectorial angles are in arithmetical progression, are them- 
selves in geometrical progression; and (2) that the tangent at any 
point of a logarithmic spiral makes a constant 
angle (called the angle of the spiral) with the 
polar radius vector. 

The former of these two propositions may be Q 
written in what is, perhaps, a simpler form, as 
follows: radii which form equal angles about the 
pole of the logarithmic spiral, are themselves / \ 
continued proportionals. That is to say, in \ 
Fig. 261, when the angle ROQ is equal to the \ 
angle MOP, then OR : OQ: : OQ: OP. \ 

A particular case of this proposition is when \ 
the equal angles are each angles of 360°: that is 
to say when in each case the radius vector makes SBS ae 
a complete revolution, and when, therefore P, Q Fig. 261. 
and R all he upon the same radius. 


4 
4 


R. 
fo 


518 THE LOGARITHMIC SPIRAL [CH. 


It was by observing, with the help of very careful measure- 
ment, this continued proportionality, that Moseley was enabled 
to verify his first assumption, based on the general appearance of 
the shell, that the shell of Nautilus was actually a logarithmic 
spiral, and this demonstration he was immediately afterwards 
in a position to generalise by extending it to all the spiral 
Ammonitoid and Gastropod mollusca*. 

For, taking a median transverse section of a Nautilus pompilius, 
and carefully measuring the successive breadths of the whorls 
(from the dark line which marks what was originally the outer 
surface, before it was covered up by fresh deposits on the part 
of the growing and advancing shell), Moseley found that “the 
distance of any two of its whorls measured upon a radius vector 
is one-third that of the two next whorls measured upon the same 
radius vector}. Thus (in Fig. 262), ab is one-third of bce, de of 
ef, gh of hi, and kl of Im. The curve is therefore a logarithmic 
spiral.” 

The numerical ratio in the case of the Nautilus happens to 
be one of unusual simplicity. Let us take, with Moseley, a 
somewhat more complicated example. 

From the apex of a large specimen of Turbo duplicatust a 

* The Rev. H. Moseley, On the Geometrical Forms of Turbinated and Discoid 
Shells, Phil. Trans. pp. 351-370, 1838. 

+ It will be observed that here Moseley, speaking a8 a mathematician and 
considering the linear spiral, speaks of whorls when he means the linear boundaries, 
or lines traced by the revolving radius vector; while the conchologist usually 
applies the term whorl to the whole space between the two boundaries. As con- 
chologists, therefore, we call the breadth of a whorl what Moseley looked upon as 
the distance between two consecutive whorls. But this latter nomenclature Moseley 
himself often uses. 

t In the case of Turbo, and all other “turbinate” shells, we are dealing not with 
a plane logarithmic spiral, as in Nautilus, but with a ‘‘ gauche” spiral, such 
that the radius vector no longer revolves in a plane perpendicular to the axis of 
the system, but is inclined to that axis at some constant angle (@). The figure 
still preserves its continued similarity, and may with strict accuracy be called a 
logarithmic spiral in space. It is evident that its envelope will be a right circular 
cone; and ind:ed it is commonly spoken of as a logarithmic spiral wrapped wpon 
a cone, its pole coinciding with the apex of the cone. It follows that the distances 
of successive whorls of the spiral measured on the same straight line passing through 


the apex of the cone, are in geometrical pregression, and conversely: just as in 
the former case. But the ratio between any two consecutive interspaces (i.e. 


R, — R,/R, — R,) is now equal to 2788 Cot 4 being the semi-angle of the enveloping 
cone. (Cf. Moseley, Phil. Mag. xx1, p. 300, 1842.) 


x1] ; OF THE NAUTILUS SHELL 519 


line was drawn across its whorls, and their widths were measured 
upon it in succession, beginning with the last but one. The 
measurements were, as before, made with a fine pair of compasses 
and a diagonal scale. The sight was assisted by a magnifying 
glass. In a parallel column to the following admeasurements 
are the terms of a geometric progression, whose first term is the 
width of the widest whorl measured, and whose common ratio is 
1-1804. 


Fig. 262. 
Widths of successive Terms of a geometrical progression, 
whorls measured in inches whose first term is the width of 
and parts of an inch the widest whorl, and whose 
common ratio is }-1804 
1-31 131 
1-12 1-1098 
94 -94018 
80 -79651 
‘67 “67476 
‘57 -57164 
“48 -48427 
“41 -41026 


The close coincidence between the observed and the calculated 
figures is very remarkable, and is amply sufficient to justify the 
conclusion that we are here dealing with a true logarithmic 
spiral. 

Nevertheless, in order to verify his conclusion still further, 
and to get partially rid of the inaccuracies due to successive small 


520 THE LOGARITHMIC SPIRAL By fs 


measurements, Moseley proceeded to investigate the same shell, 
measuring not single whorls, but groups of whorls, taken several 
at a time: making use of the following property of a geometrical 
progression, that “if yz represent the ratio of the sum of every 
even number (m) of its terms to the sum of half that number of 
terms, then the common ratio (r) of the series is represented by 


the formula 
2 


,= (pe LG ee 


Accordingly, Moseley made the following measurements, 
beginning from the second and third whorls respectively : 


Width of 

ee 
Six whorls Three whorls Ratio u 
5:37 2-03 2-645 
4-55 1-72 2-645 

Four whorls Two whorls 
4-15 1-74 2-385 
3°52 1:47 2-394 


“By the ratios of the two first admeasurements, the formula 
gives 


r = (1-645)* = 1-1804. 


By the mean of the ratios deduced from the second two admeasure- 
ments, it gives 


r = (1-389)? = 1-1806. 


“Tt is scarcely possible to imagine a more accurate verification 
than is deduced from these larger admeasurements, and we may 
with safety annex to the species Turbo duplicatus the character- 
istic number 1-18.” 

By similar and equally concordant observations, Moseley found 
for Turbo phasianus the characteristic ratio, 1-75: and for Bucci- 
num subulatum that of 1-13. 

From the table referring to Turbo duplicatus, on page 519, it 
is perhaps worth while to illustrate the logarithmic statement of 
the same facts: that is to say, the elementary corollary to the 
fact that the successive radii are in geometric progression, that 
their logarithms differ from one another by a constant amount. 


x1] OF CERTAIN OPERCULA 521 


Turbo duplicatus. 


Relative widths of Logarithms of Difference of 
successive whorls successive whorls —_ successive logarithms 
131 2-11727 —- 
112 2-04922 06805 
94 1-97313 ‘07609 
80 1-90309 07004 
67 1-82607 ‘07702 
57 1-75587 -07020 
48 1-68124 07463 


41 1-161278 -06846 
Mean difference :07207 


And -07207 is the logarithm of 1-1805. 


Fig. 263. Operculum of Turbo. 


The logarithmic spiral is not only very beautifully manifested 
in the molluscan shell, but also, in certain cases, in the httle lid 
or “operculum” by which the entrance to the tubular shell is 
closed after the animal has withdrawn itself within. In the spiral 
shell of Turbo, for instance, the operculum is a thick calcareous 
structure, with a beautifully curved outline, which grows by 
successive increments applied to one portion of its edge, and shews, 
accordingly, a spiral line of growth upon its surface. The succes- 
sive increments leave their traces on the surface of the operculum 


522 THE LOGARITHMIC SPIRAL [CH. 


(Fig. 264, 1), which traces have the form of curved lines in 
Turbo, and of straight lines in (e.g.) Nerita (Fig. 264, 2); that 
is to say, apart from the side constituting the outer edge of the 
operculum (which side is always and of necessity curved) the 
successive increments constitute curvilinear triangles in the one 
case, and rectilinear triangles in the other. The sides of these 
triangles are tangents to the spiral line of the operculum, and 
may be supposed to generate it by their consecutive intersections. 

In a number of such opercula, Moseley measured the breadths 
of the successive whorls along a radius vector*, just in the same 


Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After Moseley.) 


way as he did with the entire shell in the foregoing cases; and 
here is one example of his results. 


Operculum of Turbo sp.; breadth (an inches) of successive 
whorls, measured from the pole. 


Distance Ratio Distance Ratio Distance Ratio Distanee Ratio 

24 “16 2 18 
2-28 2-31 2-30 2°30 

915) 3317 6 “42 
2-o2 2-30 2°30 2-24 

1-28 85 1:38 94 


* As the successive increments evidently constitute similar figures, similarly 
related to the pole (P), it follows that their linear dimensions are to one another 
as the radii vectores drawn to similar points in them: for instance as PP,, PP,, 
which (in Fig. 264, 1) are radii vectores drawn to the points where they meet the 
common boundary. 


x1] OF CERTAIN OPERCULA 523 


The ratio is approximately constant, and this spiral also is, | 
therefore, a logarithmic spiral. 

But here comes in a very beautiful illustration of that property 
of the logarithmic spiral which causes its whole shape to remain 
unchanged, in spite of its apparently unsymmetrical, or unilateral, 
mode of growth. For the mouth of the tubular shell, into which 
the operculum has to fit, is growing or widening on all sides: 
while the operculum is increasing, not by additions made at the 
same time all round its margin, but by additions made only on 
one side of it at each successive stage. One edge of the operculum 
thus remains unaltered as it is advanced into each new position, 
and as it is placed in a newly formed section of the tube, similar 
to but greater than the last. Nevertheless, the two apposed 
structures, the chamber and its plug, at all times fit one another 
to perfection. The mechanical problem (by no means an easy 
one), is thus solved: “How to shape a tube of a variable section, 
so that a piston driven along it shall, by one side of its margin, 
coincide continually with its surface as it advances, provided only 
that the piston be made at the same time continually to revolve 
in its own plane.” 

As Moseley puts it: “That the same edge which fitted a portion 
of the first less section should be capable of adjustment, so as to 
fit a portion of the next similar but greater section, supposes 
a geometrical provision in the curved form of the chamber of 
great apparent complication and difficulty. But God hath 
bestowed upon this humble architect the practical skill of a 
learned geometrician, and he makes this provision with admirable 
precision in that curvature of the logarithmic spiral which he 
gives to the section of fhe shell. This curvature obtaining, he 
has only to turn his operculum slightly round in its own plane as 
he advances it into each newly formed portion of his chamber, 
to adapt one margin of it to a new and larger surface and a different 
curvature, leaving the space to be filled up by increasing the 
operculum wholly on the other margin.” 

But in many, and indeed more numerous Gastropod mollusca, 
the operculum does not grow in this remarkable spiral fashion, 
but by the apparently much simpler process of accretion by 
concentric rings. This suggests to us another mathematical 


524 THE LOGARITHMIC SPIRAL [cH. 


feature of the logarithmic spiral. We have already seen that the 
logarithmic spiral has a number of “limiting cases,” apparently 
very diverse from one another. Thus the right cone is a logarith- 
mic spiral in which the revolution of the radius vector is infinitely 
slow; and, in the same sense, the straight line itself is a hmiting 
case of the logarithmic spiral. The spiral of Archimedes, though 
not a limiting case of the logarithmic spiral, closely resembles 
one in which the angle of the spiral is very near to 90°, and the 
spiral is coiled around a central core. But if the angle of the 
spiral were actually 90°, the radius vector would describe a circle, 
identical with the “core” of which we have just spoken; and 
accordingly it may be said that the circle is, in this sense, a true 
limiting case of the logarithmic spiral. In this sense, then, the 
circular concentric operculum, for instance of Turritella or 
Littorina, does not represent a breach of continuity, but a “limiting 
case”’ of the spiral operculum of Turbo; the successive “ gnomons”’ 
are now not lateral or terminal additions, but complete concentric 
rings. 


Viewed in regard to its own fundamental properties and to 
those of its limiting cases, the logarithmic spiral is the simplest 
of all known curves; and the rigid uniformity of the simple laws, 
or forces, by which it is developed sufficiently account for its 
frequent manifestation in the structures built up by the slow and 
steady growth of organisms. . 

In order to translate into precise terms the whole form and 
growth of a spiral shell, we should have to employ a mathematical 
notation, considerably more complicated than any that I have 
attempted to make use of in this book. But, in the most ele- 
mentary language, we may now at least attempt to describe the 
general method, and some of the variations, of the mathematical 
development of the shell. . 

Let us imagine a closed curve in space, whether circular or 
elliptical or of some other and more complex specific form, not 
necessarily in a plane: such a curve as we see before us when we 
consider the mouth, or terminal orifice, of our tubular shell; and 
let us imagine some one characteristic point within this closed 
curve, such as its centre of gravity. Then, starting from a fixed 


x1] OF THE MOLLUSCAN SHELL 525 


= 


origin, let this centre of gravity describe an equiangular spiral in 
space, about a fixed axis (namely the axis of the shell), while at 
the same time the generating curve grows, with each angular 
increment of rotation, in such a way as to preserve the symmetry 
of the entire figure, with or without a simultaneous movement 
of translation along the axis. 

It is plain that the entire resulting shell may now be looked 
upon in either of two ways. It is, on the one hand, an ensemble 


Fig. 265. Melo ethiopicus, L. 


of similar closed curves spirally arranged in space, gradually’ in- 
creasing in dimensions, in proportion to the increase of their 
vectorial angle from the pole. In other words, we can imagine 
our shell cut up into a system of rings, following one another in 
continuous spiral succession from that terminal and largest one, 
which constitutes the lip of the orifice of the shell. Or, on the 
other hand, we may figure to ourselves the whole shell as made 
up of an ensemble of spiral lines in space, each spiral having been 


526 THE LOGARITHMIC SPIRAL [CH. 


traced out by the gradual growth and revolution of a radius 
vector from the pole to a given point of the generating curve. 
Both systems of lines, the generating spirals (as these latter 
may be called), and the closed generating curves corresponding 
to successive margins or lips of the shell, may be easily traced 
in a great variety of cases. Thus, for example, in Dolium, 
Eburnea, and a host of others, the generating spirals are beautifully 


Fig. 266. 1, Harpa; 2, Dolium. The ridges on the shell correspond 


a 


in (1) to generating curves, in (2) to generating spirals. 


marked out by ridges, tubercles or bands of colour. In Trophon, 
Scalaria, and (among countless others) in the Ammonites, it is 
the successive generating curves which more conspicuously leave 
their impress on the shell. And in not a few cases, as in 
Harpa, Dolium perdix, etc., both alike are conspicuous, ridges 
and colour-bands intersecting one another in a beautiful isogonal 


system. 


x1] OF THE MOLLUSCAN SHELL 527 


In the complete mathematical formula (such as I have not 
ventured to set forth*) for any given turbinate shell, we should 
have, accordingly, to include factors for at least the following 
elements: (1) for the specific form of the section of the tube, 
which we have called the generating curve; (2) for the specific 
rate of growth of this generating curve; (3) for its specific rate 
of angular rotation about the pole, perpendicular to the axis; 
(4) in turbinate (as opposed to nautiloid) shells, for its rate of 
shear, or screw-translation parallel to the axis. There are also 
other factors of which we should have to take account, and which 
would help to make our whole expression a very complicated one. 
We should find, for instance, (5) that in very many cases our 
generating curve was not a plane curve, but a sinuous curve in 
three dimensions; and we should also have to take account 
(6) of the inclination of the plane of this generating curve to the 
axis, a factor which will have a very important influence on the 
form and appearance of the shell. For instance in Haliotis it is 
obvious that the generating curve lies in a plane very oblique to 
the axis of the shell. Lastly, we at once perceive that the ratios 
which happen to exist between these various factors, the ratio 
for instance between the growth-factor and the rate of angular 
revolution, will give us endless possibilities of permutation of 
form: For instance (7) with a given velocity of vectorial rotation, 
a certain rate of growth in the generating curve will give us a 
spiral shell of which each successive whorl will just touch its 
predecessor and no more; with a slower growth-factor, the whorls 
will stand asunder, as in a ram’s horn; with a quicker growth- 
factor, each whorl will cut or intersect its predecessor, as in an 
Ammonite or the majority of gastropods, and so on (cf. p. 541). 

In like manner (8) the ratio between the growth-factor and 
the rate of screw-translation parallel to the axis will determine 
the apical angle of the resulting conical structure: will give us 
the difference, for example, between the sharp, pointed cone of 
Turritella, the less acute one of Fusus or Buccinum, and the 


* The equation to the surface of a turbinate shell is discussed by Moseley 
(Phil. Trans. tom. cit. p. 370), both in terms of polar coordinates and of the rect- 
angular coordinates +, y, z. A more elegant representation can be given in vector 
notation, by the method of quaternions. 


528 THE LOGARITHMIC SPIRAL [CH. 


obtuse one of Harpa or Dolium. In short it is obvious that all 
the differences of form which we observe between one shell and 
another are referable to matters of degree, depending, one and all, 
upon the relative magnitudes of the various factors in the complex 
equation to the curve. 


The paper in which, nearly eighty years ago, Canon Moseley 
thus gave a simple mathematical expression to the spiral forms of 
univalve shells, is one of the classics of Natural History. But 
other students before him had come very near to recognising 
this mathematical simplicity of form and structure. About the 
year 1818, Reinecke had suggested that the relative breadths of 
the adjacent whorls in an Ammonite formed a constant and 
diagnostic character; and Leopold von Buch accepted and 
developed the idea*. But long before, Swammerdam, with a 
deeper insight, had grasped the root of the whole matter: for, 
taking a few diverse examples, such as Helix and Spirula, he 
shewed that they and all other spiral shells whatsoever were 
referable to one common type, namely to that of a simple tube, 
variously curved according to definite mathematical laws; that 
all manner of ornamentation, in the way of spines, tuberosities, 
colour-bands and so forth, might be superposed upon them, but 
the type:was one throughout, and specific differences were of a 
geometrical kind. “Omnis enim quae inter eas animadvertitur 
differentia ex sola nascitur diversitate gyrationum: quibus si 
insuper externa quaedam adjunguntur ornamenta pinnarum, 
sinuum, anfractuum, planitierum, eminentiarum, profunditatum, 
extensionum, impressionum, circumvolutionum, colorumque:... 
tune deinceps facile est, quarumcumque Cochlearum figuras 
geometricas, curvosque, obliquos atque rectos angulos, ad unicam 
omnes speciem redigere: ad oblongum videlicet tubulum, qui 
vario modo curvatus, crispatus, extrorsum et introrsum flexus, 
ita concrevit Tt.” 

* J. C. M. Reinecke, Maris protogaet Nautilos, etc., Coburg. 1818. Leopold 
von -Buch, Ueber die Ammoniten in den Alteren Gebirgsschichten, Abh. Berlin. 
Akad., Phys. Kl. pp. 135-158, 1830; Ann. Sc. Nat. xxvmt, pp. 5-43, 1833; cf. 
Elie de Beaumont, Sur lenroulement des Ammonites, Soc. Philom., Pr. verb. 


pp. 45-48, 1841. 
+ Biblia Naturae sive Historia Insectorum, Leydae, 1737, p. 152. 


XI] OF THE MOLLUSCAN SHELL 529 


For some years after the appearance of Moseley’s paper, a 


number of writers followed in _ his 
footsteps, and attempted, in various 
ways, to put his conclusions to 
practical use. Forinstance, D’Orbigny 
devised a very simple protractor, which 
he called a Helicometer*, and which 
is represented in Fig. 267. By means 
of this little instrument, the apical 
angle of the turbinate shell was im- 
mediately read off, and could then 
be used as a specific and diagnostic 
character. By keeping one lhmb of 
the protractor parallel to the side of 
the cone while the other was brought 
into line with the suture between two 
adjacent whorls, another specific angle, 
the “sutural angle,’ could in like 
manner be recorded. And, by the 
linear scale upon the instrument, the 
relative breadths of the consecutive 
whorls, and that of the terminal 
chamber to the rest of the shell, 
might also, though somewhat roughly, 
be determined. For instance, in 
Terebra dimidiata, the apical angle 
was found to be 13°, the sutural 
angle 109°, and so forth. 

It was at once obvious that, in 
such a shell as is represented in 
Fig. 267 the entire outline of the 


Fig. 267. D’Orbigny’s 
Helicometer. 


shell (always excepting that of the immediate neighbourhood of 


* Alcide D’Orbigny, Bull. de la soc. géol. Fr. xm, p. 200, 1842; Cours élém. 
de Paléontologie, 1, p. 5, 1851. A somewhat similar instrument was described by 
Boubée. in Bull. soc. géol. 1, p. 232, 1831. Naumann’s Conchyliometer (Poggend. 
Ann. LIV, p. 544, 1845) was an application of the screw-micrometer; it was provided 
also with a rotating stage, for angular measurement. It was adapted for the 
study of a discoid or ammonitoid shell, while D’Orbigny’s instrument was meant 


for the study of a turbinate shell. 


Tr -Ge 


o+ 


530 THE LOGARITHMIC SPIRAL [on. 


the mouth) could be restored from a broken fragment. For if we 
draw our tangents to the cone, it follows from the symmetry 
of the figure that we can continue the -projection of the sutural 
line, and so mark off the successive whorls, by simply drawing 
a series of consecutive parallels, and by then filling into the 
quadrilaterals so marked off a series of curves similar to one 
another, and to the whorls which are still intact in the broken 
shell. 

But the use of the helicometer soon shewed that it was by no 
means universally the case that one and the same right cone was 
tangent to all the turbinate whorls; in other words, there was not 
always one specific apical angle which held good for the entire 
system. In the great majority of cases, it is true, the same 
tangent touches all the whorls, and is a straight line. But in 
others, as in the large Cerithium nodosum, such a line is slightly 
convex to the axis of the shell; and in the short spire of Dohum, 
for instance, the convexity is marked, and the apex of the spire 
is a distinct cusp. On the other hand, in Pupa and Clausilia, the 
common tangent is concave to the axis of the shell. 

So also is it, as we shall presently see, among the Ammonites: 
where there are some species in which the ratio of whorl to whorl 
remains, to all appearance, perfectly constant; others in which 
it gradually, though only slightly increases; and others again in 
which it slightly and gradually falls away. It is obvious that, 
among the manifold possibilities of growth, such conditions as 
these are very easily conceivable. It is much more remarkable 
that, among these shells, the relative velocities of growth in various 
dimensions should be as constant as it is, than that there should 
be an occasional departure from perfect regularity. In such cases 
as these latter, the logarithmic law of growth is only approximately 
true. The shell is no longer to be represented as a right cone 
which has been rolled up, but as a cone which had grown trumpet- 
shaped, or conversely whose mouth had narrowed in, and which 
in section is a curvilinear instead of a rectilinear triangle. But 
all that has happened is that a new factor, usually of small or all 
but imperceptible magnitude, has been introduced into the case; 
so that the ratio, log r = @ log a, is no longer constant, but varies 
slightly, and in accordance with some simple law. 


XI] OF NAUMANN’S CONCHOSPIRAL — 531 


Some writers, such as Naumann and Grabau, maintained that 
the molluscan spiral was no true logarithmic spiral, but differed 
from it specifically, and they gave to it the name of Conchosyiral. 
They pointed out that the logarithmic spiral originates in a 
mathematical point, while the molluscan shell starts with a little 
embryonic shell, or central chamber (the “protoconch” of the 
conchologists), around which the spiral is subsequently wrapped. 
It is plain that this undoubted and obvious. fact need not 
affect the logarithmic law of the shell as a whole; we have 
only to add a small constant to our equation, which becomes 
r=m+a’. 

There would seem, by the way, to be considerable confusion 
in the books with regard to the so-called “protoconch.” In many 
cases it is a definite®tructure, of simple form, representing the 
more or less globular embryonic shell before it began to elongate 
into its conical or spiralform. But in many cases what is described 
as the “protoconch” is merely an empty space in the middle of 
the spiral coil, resulting from the fact that the actual spiral shell 
has a definite magnitude to begin with, and that we cannot follow 
it down to its vanishing point in infinity. For instance, in the 
accompanying figure, the large space a 
is stvled the protoconch, but it is the 
little bulbous or hemispherical chamber 
within it, at the end of the spire, 
which is the real beginning of the 
tubular shell. The form and magni- 
tude of the space a are determined by 
the “angle of retardation,” or ratio of 
rate of growth between the inner and 
outer curves of the spiral shell. They 
are independent of the shape and size of 
the embryo, and depend only (as we shall 
see better presently) on the direction and relative rate of growth 
of the double contour of the shell. 


Now that we have dealt, in a very general way, with some of 
the more obvious properties of the logarithmic spiral, let us 
consider certain of them a little more particularly, keeping in 


34—2 


532 THE LOGARITHMIC SPIRAL [CH. 


view as our chief object the investigation (on elementary lines) 
of the possible manner and range of variation of the molluscan 
shell. 

There is yet another equation to the logarithmic spiral, 
very commonly employed, and without the. 
help of which we shall find that we cannot 
get far. It is as follows: 


ea e2cota 


This follows directly from the fact that 
the angle a (the angle between the radius 
vector and the tangent to the curve) is 
constant. 

For, then, 


tana (= tan d) = rdé/dr, 
Fig. 269. therefore dr/r = d@ cot a, 


and, integrating, 
log r = 6 cota, 


or é pe ote 


As we have seen throughout our preliminary discussion, the 
two most important constants (or chief “specific characters,” as 
the naturalist would say) in any given logarithmic spiral, are 
(1) the magnitude of the angle of the spiral, or “constant angle,” 
a, and (2) the rate of increase of the radius vector for any given 
angle of revolution, 6. Of this latter, the simplest case is when 

= 27, or 360°; that is to say when we compare the breadths, 
along the same radius vector, of two successive whorls. As our 
two magnitudes, that of the constant angle, and that of the ratio 
of the radu or breadths of whorl, are related to one another, we 
may determine either of them by actual measurement and proceed 
to calculate the other. 

In any complete spiral, such as that of Nautilus, it is (as we 
have seen) easy to measure any two radu (r), or the breadths in 


x1] ITS MATHEMATICAL PROPERTIES 533 


a radial direction of any two whorls (W). We have then merely 
to apply the formula 


= } _— ,820ta = ¢cot 
P| as sane » Or W accal Wx =e ¢ 


which we may simply write 7 = e°°°**, etc.; since our first radius 
or whorl is regarded, for the purpose of comparison, as being equal 
to unity. 

Thus, in the diagram, OC/OE, or EF/BD, or DC/EF, being 
in each case radii, or diameters, at right angles to one another, 


are all equal to piven While in like manner, HO/OF, EG/FH, 
or GO/HO, all equal e7°°t*; and BC/BA, or CO/OB = e?"°"*. 


Fig. 270. 


As soon, then,-as we have prepared tables for these values, 
the determination of the constant angle a in a particular shell 
becomes a very simple matter. 

A complete table would be cumbrous, and it will be sufficient 
to deal with the simple case of the ratio between the breadths of 
adjacent, or immediately succeeding, whorls. 

Here we have r =e7""*, or logr = loge x 27 x cota, from 
which we obtain the following figures * : 


* It is obvious that the ratios of opposite whorls, or of radii 180° apart, are 
represented by the square roots of these values; and the ratios of whorls or radii 
90° apart, by the square roots of these again. 


534 THE LOGARITHMIC SPIRAL [CH. 


Ratio of breadth of each 


whorl to the next preceding Constant angle 

r/1 a 

i-1 Bon ao 

1-25 87 58 

1-5 86 18 

2-0 83 42 

2°5 81 42 

3°0 80 5 

35 78 43 

4-0 77 34 

4:5 76 32 

5:0 75 38 

10-0 69 53 

20-0 64 31 

50-0 58 5 

100-0 53 46 

1000-0 42 17 : 

10,000 34 19 

100,000 28 37 

1,000,000 24 28 

10,000,000 21 18 ty 
100,000,000 18 50 
1,000,000,000 16 52 


We learn several interesting things from this short table. We 
see, in the first place, that where each whorl is about three times 
the breadth of its neighbour and predecessor, as is the case in 
Nautilus, the constant angle is in the neighbourhood of 80°; and 
hence also that, in all the ordinary Ammonitoid shells, and in all 
the typically spiral shells of the Gastropods*, the constant angle 


S00 


Fig. 271. 


is also a large one, being very seldom less 
than 80°, and usually between 80° and 
85°. In the next place, we see that with 
smaller angles the apparent form of the 
spiral is greatly altered, and the very fact 
of its being a spiral soon ceases to be 
apparent (Figs. 271, 272). Suppose one 
whorl to be an inch in breadth, then, if 
the angle of the spiral were 80°, the 


* For the correction. to be applied in the case of the helicoid, or “‘turbinate” 


shells, see p. 557. 


x1] ITS MATHEMATICAL PROPERTIES 5BE 


U 


~ 


next whorl would (as we have just seen) be about three inches 
broad; if it were 70°, the next whorl would be nearly ten inches, 
and if it were 60°, the next whorl would be nearly four feet 
broad. If the angle were 28°, the next whorl would be a mile 
‘and a half in breadth; and if it were 17°, the next would be 
some 15,000 miles broad. 


60° 


ay, 


Fig. 272. 


In other words, the spiral shells of gentle curvature, or of 
small constant angle, such as Dentalium or Nodosaria, are true 
logarithmic spirals, just as are those of Nautilus or Rotalia: 
from which they differ only in degree, in the magnitude of an 
angular constant. But this diminished magnitude of the angle 
causes the spiral to dilate with such immense rapidity that, so 
to speak, “it never comes round”; and so, in such a shell as 
Dentalium, we never see but a small portion of the initial whorl. 


Fig. 273. 


We might perhaps be inclined to suppose that, in such a shell as Dentalium, 
the lack of a visible spiral convolution was only due to our seeing but a small 
portion of the curve, at a distance from the pole, and when, therefore, its 


536 THE LOGARITHMIC SPIRAL [CH. 


curvature had already greatly diminished. That is to say we might suppose 
that, however small the angle a, and however rapidly the whorls accordingly 
increased, there would nevertheless be a manifest spiral convolution in the 
immediate neighbourhood of the pole, as the starting point of the curve. 
But it may be shewn that this is not so. 


For, taking the formula r = ae’ Ob, 
this, for any given spiral, is equivalent to ae? 
Therefore log (r/a) = ké, 
or, I/k = =a : 

** Tog (r/a) 


Then, if 6 increase by 27, while r increases to 7, 


iy 6427 
k log (ry/a)’ 


which leads, by subtraction to 
L/k . log (7,/r) = 27. 
Now, as a tends to 0, k (i.e. cot a) tends to #, and therefore, as k —> o, 
log (r,/r) —~> and also 7r,/r —> o. 
Therefore if one whorl exists, the radius vector of the other is infinite; 
in other words, there is nowhere, even in the near neighbourhood of the 
pole, a complete revolution of the spire. Our spiral shells of small constant 


angle, such as Dentalium, may accordingly be considered to represent suf- 
ficiently well the true commencement of their respective spirals. 


Let us return to the problem of how to ascertain, by direct 
measurement, the spiral angle of any particular shell. The 
method already employed is only applicable to complete spirals, 
that is to say to those in which the angle of the spiral is large, 
and furthermore it is inapplicable to portions, or broken fragments, 
of a shell. In the case of the broken fragment, it is plain that the 
determination of the angle is not merely of theoretic interest, 
but may be of great practical use to the conchologist as being the 
one and only way by which he may restore the outline of the 
missing portions. We have a considerable choice of methods, 
which have been summarised by, and are partly due to, a very 
careful student of the Cephalopoda, the late Rev. J. F. Blake*. 


* On the Measurement of the Curves formed by Cephalopods and other Mollusks 
Phil. Mag. (5), V1, pp. 241-263, 1878. 


x1] ITS MATHEMATICAL PROPERTIES 537 


(1) The following method is useful and easy when we have 
a portion of a single whorl, such as to shew both its inner and its 
outer edge. A broken whorl of an Ammonite, a 
curved shell such as Dentalium, or a horn of 
similar form to the latter, will fall under this 
head. We have merely to draw a tangent, 
GEH, to the outer whorl at any point #; then 
draw to the inner whorl a tangent parallel to 
GEH, touching the curve in some point Ff. The 
straight line joining the points of contact, HF, 
must evidently pass through the pole: and, 
accordingly, the angle GHF is the angle re- 
quired. In shells which bear longitudinal striae 
or other ornaments, any pair of these will 
suffice for our purpose, instead of the actual Fig. 274. 
boundaries of the whorl. But it is obvious that 
this method will be apt to fail us when the angle a is very small; 
and when, consequently, the points H and F are very remote. 

(2) In shells (or horns) shewing rings, or other transverse 
ornamentation, we may take it that these ornaments are set at 
a constant angle to the spire, and therefore to the radi. The angle 
(?) between two of them, as AC, BD, is therefore equal to the 


Fig. 275. An Ammonite, to 
shew corrugated surface- 
pattern. 


angle @ between the polar radii from A and B, or from C and D; 
and therefore BD/AC = e°°°**, which gives us the angle a in terms 
of known quantities. 


538 THE LOGARITHMIC SPIRAL [CH. 


(3) If only the outer edge be available, we have the ordinary 
geometrical problem,—given an arc of an equiangular spiral, to 
find its pole and spiral angle. The methods we may employ 
depend (1) on determining directly the position of the pole, and 
(2) on determining the radius of curvature. 

The first method is theoreti- 
cally simple, but difficult im 
practice; for it requires great 
accuracy in determining the 
points. Let AD, DB, be two 
tangents drawn to the curve. 
Then a circle drawn through the 
points ABD will pass through 
the pole O; since the angles OAD, 
OBE (the supplement of OBD), 
are equal. The point,O may be 
determined by the intersection of two such circles; and the angle 
DBO is then the angle, a, required. 

Or we may determine, graphically, at two points, the radii of | 
curvature, p,p2. Then, if.s be the length of the arc between them 
(which may be determined with fair accuracy by rolling the margin 
of the shell along a ruler) 


cot a = (py — ps)/s. 


The following method*, given by Blake, will save actual determination of 
the radii of curvature. 

Measure along a tangent to the curve, the distance, AC, at which a certain 
small offset, CD, is made by the curve; and from another point B, measure 
the distance at which the curve makes an equal offset. Then, calling the 
‘offset wp; the arc AB, s; and AC, BE, respectively x,, x,, we have 


Fig. 277. 


2 : 
= Gis SHS , approximately, 
2h 
and cot a = Was ee) 
2s 


Of all these methods by which the mathematical constants, 
or specific characters, of a given spiral shell may be determined, 
the only one of which much use has been made is that which 
Moseley first employed, namely, the simple method of determining 


* For an example of this method, see Blake, /.c. p. 251. 


XI] IN CERTAIN AMMONITES 539 


the relative breadths of the whorl at distances separated by some 
convenient vectorial angle (such as 90°, 180°, or 360°). 

Very elaborate measurements of a number of Ammonites have 
been made by Naumann*, by Sandbergert, and by Grabaut, 
among which we may choose a couple of cases for consideration. 
In the following table I have taken a portion of Grabau’s deter- 
minations of the breadth of the whorls in Ammonites (Arcestes) 


Ammonites intuslabiatus. 


Ratio of breadth of 
Breadth of whorls successive whorls ‘The angle (a) 


(180° apart) (360° apart) as calculated 
0-30 mm. - — — 
0-30 1-333 87-23" 
0-40 1-500 86 19 
0-45 1-500 86 19 
0-60 1-444 86 39 
0-65 1-417 86 49 
0-85 1-692 85 13 
1-10 1-588 85 47 
1-35 1-545 86 2 
1-70 1-630 85 33 
2-20 1-441 86 40 
2-45 1-432 86 43 
35 2 7/ahs 8 0 
4-25 1-683 85 16 
5-30 1-482 86 25 
6-30 1-519 86 12 
8-05 1-635 85 32 
10-30 1-416 86 50 
11-40 1-252 F 87 57 
12-90 — —_— — 


Mean 86° 15’ 


* Naumann, C. F., Ueber die Spiralen von Conchylien, Abh. k. sdichs. Ges. pp. 153— 
196, 1846; Ueber die cyclocentrische Conchospirale u. iiber das Windungsgesetz von 
Planorbis corneus, ibid. 1, pp. 171-195, 1849; Spirale von Nautilus u. Ammonites 
galeatus, Ber. k. sdchs. Ges. 1, p. 26, 1848; Spirale von Amm. Ramsaueri, ibid. Xv1, 
p- 21, 1864; see also Poggendorff’s Annalen, L, p. 223, 1840; Li, p. 245, 1841; Liv, 
p. 541, 1845, ete. ; 

+ Sandberger, G., Spiralen des Ammonites Amaltheus, A. Gaytani, und Goniatites 
intumescens, Zeitschr. d. d. Geol. Gesellsch. x, pp. 446-449, 1858. 

t Grabau, A. H.. Ueber die Nawmannsche Conchospirale, etc. Inauguraldiss. 
Leipzig, 1872: Die Spiralen von Conchylien, etc. Programm, Nr. 502, Leipzig, 
1882. 


540 THE LOGARITHMIC SPIRAL [CH. 


intuslabiatus ; these measurements Grabau gives for every 45° of 
arc, but I have only set forth one quarter of these measurements, 
that is to say, the breadths of successive whorls measured along 
one diameter on both sides of the pole. The ratio between 
alternate measurements is therefore the same ratio as Moseley 
adopted, namely the ratio of breadth between contiguous whorls 
along a radius vector. I have then added to these observed 
values the corresponding calculated values of the angle a, as 
obtained from our usual formula. 

There is considerable irregularity in the ratios derived from 
these measurements, but it will be seen that this irregularity only 
imphes a variation of the angle of the spiral between about 85° 
and 87°; and the values fluctuate pretty regularly about the 
mean, which is 86° 15’. Considering the difficulty of measuring 
the whorls, especially towards the centre, and in particular the 
difficulty of determining with precise accuracy the position of the 
pole, it is clear that in such a case as this we are scarcely justified 
in asserting that the law of the logarithmic spiral.is departed from. 

In some cases, however, it is undoubtedly departed from. 
Here for instance is another table from Grabau, shewing the 
corresponding ratios in an Ammonite of the group of Arcestes 
tornatus. In this case we see a distinct tendency of the ratios to 


Ammonites tornatus. 


Ratio of breadth of — The spiral 


Breadth of whorls successive whorls angle (a) as 
(180° apart) (360° apart) calculated 
0-25 mm. — — 
0-30 1-400 86° 56’ 
0°35 1-667 85 21 
0-50 2-000 83 42 
0:70 2-000 83 42 
1-00 2-000 83 42 
1-40 2-100 83 16 
2-10 2-179 82 56 
b 3-05 2-238 82 42 
4-70 2-492 81 44 
7-60 2-574 81 27 
12-10 2-546 81 33 
19-35 — —_— — 


Mean 83° 22’ 


x1] OF SHELLS GENERALLY 541 


increase as we pass from the centre of the coil outwards, and 
consequently for the values of the angle a to diminish. The case 
is precisely comparable to that of a cone with slightly curving 
sides: in which, that is to say, there is a shght acceleration 
of growth in a transverse as compared with the longitudinal 
direction. 


In a tubular spiral, whether plane or helicoid, the consecutive 
whorls may either be (1) isolated and remote from one another; 
or (2) they may precisely meet, so that the outer border of one 
and the inner border of the next just coincide; or (3) they may 
overlap, the vector plane of each outer whorl cutting that of its 
immediate predecessor or predecessors. 

Looking, as we have done, upon the spiral shell as being 
essentially a cone rolled up, it is plain that, for a given spiral 
angle, intersection or non-intersection of the successive whorls 
will depend upon the agical angle of the original cone. For the 
wider the cone, the more rapidly will its inner border tend to 
encroach on the outer border of the preceding whorl. 

But it is also plain that the greater be the apical angle of the 
cone, and the broader, consequently, the cone itself be, the greater 
difference will there be between the total lengths of its inner and 
outer border, under given conditions of flexure. And, since the 
inner and outer borders are describing precisely the same spiral 
about the pole, it is plain that we may consider the inner border . 
as being retarded in growth as compared with the outer, and as 
being always identical with a smaller and earlier part of ‘the 
latter. 

If A be the ratio of growth between the outer and the inner 
curve, then, the outer curve being represented by 


Ps ae oe 


the equation to the inner one will be 


f—vaNe COP, 


—B) 5 
or ry = ae? Preot« 


and 6 may then be called the angle of retardation, to which the 
inner curve is subject by virtue of its slower rate of growth. 


542 THE LOGARITHMIC SPIRAL [CH. 


Dispensing with mathematical formulae, the several conditions 
may be illustrated as follows: 

In the diagrams (Fig. 278), OP,P,Ps;, etc. represents a radius, 
on which P,, P,, P;, are the points attained by the outer border 
of the tubular shell after as many entire consecutive revolutions. 
And P,’, P,’, P,', are the points similarly intersected by the inner 
border; OP/OP’ being always = 4, which is the ratio of growth, 
or “cutting-down factor.” Then, obviously, when OP, is less 
than OP,’ the whorls will be separated by an interspace (qa); 
(2) when OP, = OP,’ they will be in contact (6b), and (3) when 
OP, is greater than OP,’ there will a greater or less extent of 


Fig. 278. 


overlapping, that.is to say of concealment of the surfaces of the 
earlier by the later whorls (c). And as a further case (4), it is 
plain that if A be very large, that is to say if OP, be greater, not 
only than OP,’ but also than OP,', OP,', etc., we shall have 
complete, or all but complete concealment by the last formed 
whorl, of the whole of its predecessors. This latter condition 
is completely attained in Nautilus pompilius, and approached, 
though not quite attained, in NV. wmbilicatus; and the difference 
between these two forms, or “species,” 1s constituted accordingly 
by a difference in the value of A. (5) There is also a final case, 
not easily distinguishable externally from (4), where P’ lies on 


x1] OF SHELLS GENERALLY 543 


the opposite side of the radius vector to P, and is therefore 
imaginary. This final condition is exhibited in Argonauta. 

The limiting values of A are easily ascertained. 

In Fig. 279 we have portions 
of two successive whorls, whose 
corresponding points on the same 
radius vector (as R and R’) are, 
therefore, at a distance apart 
corresponding to 27. Let r and 
r’ refer to the inner, and R, R’ to 
the outer sides of the two whorls. Then, if we consider 


9 cot 
Ree ce 


4 
it follows that R’ = aei@t27) cota 


r = Aaeeot* — Gee cote 


and r = Aae@ t2n)cota _ gelO+2n—Bycot a. 


Now in the three cases (a, b, c) represented in Fig. 278, it is 
plain that 7’ 2 R, respectively. That is to say, 


ae® cot a 


IF 


rae 2) cota 


and re?" cota 


= 
=z 
= 
< 


The case in which Ae"°* = 1, or — log A = 27 cota loge, is 


the case represented in Fig. 278, b: that is to say, the particular 
case, for each value of a, where the consecutive whorls just 
touch, without interspace or overlap. For such cases, then, we 
may tabulate the values of A, as follows: 


Constant angle a Ratio (A) of rate of growth of inner border of tube, 


of spiral as compared with that of the outer border 
89° 2 896 
88 803 
87 -720 
86 -645 
85 O77 
80 -330 
75 234 
70 -1016 


65 0534 


544 THE LOGARITHMIC SPIRAL. [CH. 


We see, accordingly, that in plane spirals whose constant angle 
les, say, between 65° and 70°, we can only obtain contact between 
consecutive whorls if the rate of growth of the inner border of the 
tube be a small fraction,—a tenth or a twentieth—of that of the 
outer border. In spirals whose constant angle is 80°, contact is 
attained when the respective rates of growth are, approximately, 
as 3 to 1; while in spirals of constant angle from about 85° to 
89°, contact is attained when the rates of growth are in the ratio 
of from about 2 to “2. 


Fig. 280. 


If on the other hand we have, for any given value of a, a value 
of A greater or less than the value given in the above table, then 
we have, respectively, the conditions of separation or of overlap 
which are exemplified in Fig. 278, a and ¢. And, just as we 
have constructed this table of values-of A for the particular case 
of simple contact between the whorls, so we could construct 
similar tables for various degrees of separation, or degrees of 
overlap. 

For instance, a case which admits of simple solution is that 
in which the interspace between the whorls is everywhere a 
mean proportional between the breadths of the whorls them- 
selves (Fig. 280). 


xr] OF SHELLS GENERALLY 545 


In this case, let us call OA=R, OC =R,, and OB=r. 
We then have 
R, = OA = acct, 


R= OG = ae? ten) cate’ 
RR, pe ae? (8t7) cota = 72 oe 
And ich aa OV tae epee 
whence, equating, 1/A = e708, 
The corresponding values of 4 are as follows: 


Ratio (A) of rates of growth of outer and inner 
border, such as to produce a spiral with interspaces 
between the whorls, the breadth of which 
interspaces is a mean proportional between the 


Constant angle (a) breadths of the whorls themselves 
90° 1-00 (imaginary) 
89 95 
88 89 
87 *85 
86 “81 
85 °76 
80 57 
75 ° “43 
70 32 
: 65 -23 
60. “18 
D5 . “13 
50 -090 
45 063 
40 042 
35 026 
30 ‘016 


As regards the angle of retardation, 8, in the formula 


y’ 1 Ker cai or yr’ 2 Cork eo ae 
and in the case 
yr’ = e'r—B)cote, or — log A = (27 — 8) cota, 


* It has been pointed out to me that it does not follow at once and obviously 
that, because the interspace AB is a mean proportional between the breadths of 
the adjacent whorls, therefore the whole distance OB is a mean proportional 
between OA and OC. This is a corollary which requires to be proved; but the 
proof is easy. 


T. G. é 


co 
Ot 


546 THE LOGARITHMIC SPIRAL (CH. 


it is evident that when £ = 27, that will mean that A=1. In 
other words, the outer and inner borders of the tube are identical, 
and the tube is constituted by one continuous line. 

When A is a very small fraction, that is to say when the rates 
of growth of the two borders of the tube are very diverse, then 
6 will tend towards infinity—tend that is to say towards a con- 
dition in which the inner border of the tube never grows at all. 
This condition is not infrequently approached in nature. The 
nearly parallel-sided cone of Dentalium, or the widely separated 
whorls of Lituites, are evidently cases where A nearly approaches 
unity in the one case, and is still large in the other, 8 being 
correspondingly small; while we can easily find cases where f is 
very large, and A is a small fraction, for instance in Haliotis, or 
in Gryphaea. 

For the purposes of the morphologist, then, the main result 
of this last general investigation is to shew that all the various 
types of “open” and “closed” spirals, all the various degrees of 
separation or overlap of the successive whorls, are simply the 
outward expression of a varying ratio in the rate of growth of the 
outer as compared with the inner border of the tubular shell. 


The foregoing problem of contact, or intersection, of the suc- 
cessive whorls, is a very simple one in the case of the discoid shell 
but a more complex one in the turbinate. For in the discoid shell 
contact will evidently take place when the retardation of the 
inner as compared with the outer whorl is just 360°, and the 
shape of the whorls need not be considered. 

As the angle of retardation diminishes from 360°, the whorls 
will stand further and further apart in an open coil; as it increases 
beyond 360°, they will more and more overlap; and when the 
angle of retardation is infinite, that is to say when the true inner 
edge of the whorl does not grow at all, then the shell is said to 
be completely involute. Of this latter condition we have a 
striking example in Argonauta, and one a little more obscure in 
Nautilus pompilius. 

In the turbinate shell, the problem of contact is twofold, for 
we have to deal with the possibilities of contact on the same side 


of the axis (which is what we have dealt with in the discoid) and 
* 


x1] OF SHELLS GENERALLY 5AT 


also with the new possibility of contact or intersection on the 
opposite side; it is this latter case which will determine the 
presence or absence of an wmbilicus, and whether, if present, it 
will be an open conical space or a twisted cone. It is further 
obvious that, in the case of the turbinate, the question of contact 
or no contact will depend on the shape of the generating curve; 
and if we take the simple case where this generating curve may 
be considered as an ellipse, then contact will be found to depend 
on the angle which the major axis of this ellipse makes with the 
axis of the shell. The question becomes a complicated one, and 
the student will find it treated in Blake’s paper already referred to. 

When one whorl overlaps another, so that the generating 
curve cuts its predecessor (at a distance of 277) on the same radius 
vector, the locus of intersection will follow a spiral line upon the 
shell, which is called the “suture” by conchologists. Itis evidently 
one of that ensemble of spiral lines in space of which, as we have 
seen, the whole shell may be conceived to be constituted; and we 
might call it a “contact-spiral,” or “spiral of intersection.” In 
discoid shells, such as an Ammonite or a Planorbis, or in Nautilus 
umbilicatus, there are obviously two such contact-spirals, one on 
each side of the shell, that is to say one on each side of a plane 
perpendicular to the axis. In turbinate shells such a condition 
is also possible, but is somewhat rare. We have it for instance, 
in Solarium perspectivum, where the one contact-spiral is visible 
on the exterior of the cone, and the other hes internally, 
winding round the open cone of the umbilicus*; but this second 
contact-spiral is usually imaginary, or concealed within the 
whorls of the turbinated shell. Again, in Haliotis, one of the 
contact-spirals is non-existent, because of the extreme obliquity 
of the plane of the generating curve. In Scalaria pretiosa and 
in Spirula there is no contact-spiral, because the growth of the 
generating curve has been too slow, in comparison with the vector 
rotation of its plane. In Argonauta and in Cypraea, there is no 
contact-spiral, because the growth of the generating curve has 
been too quick. Nor, of course, is there any contact-spiral in 
Patella or in Dentalium, because the angle a is too small ever to 
give us a complete revolution of the spire. 

* A beautiful construction: stwpendum Naturae artificcwm, Linnaeus. 


35-9 


548 THE LOGARITHMIC SPIRAL [CH. 


The various forms of straight or spiral shells among the 
Cephalopods, which we have seen to be capable of complete 
definition by the help of elementary mathematics, have received 
a very complicated descriptive nomenclature from the palaeon- 
tologists. For instance, the straight cones are spoken of as 
orthoceracones or bactriticones, the loosely coiled forms as gyrocera- 
cones or mimoceracones, the more closely coiled shells, in which 
one whorl overlaps the other, as nautilicones or ammoniticones, 
and so forth. In such a succession of forms the biologist sees 
undoubted and unquestioned evidence of ancestral descent. For 
instance we read in Zittel’s Palaeontology*: ‘The bactriticone 
obviously represents the primitive or primary radical of the 
Ammonoidea, and the mimoceracone the next or secondary radical 
of this order”; while precisely the opposite conclusion was drawn 
by Owen, who supposed that the straight chambered shells of 
such fossil cephalopods as Orthoceras had been produced by the 
gradual unwinding of a coiled nautiloid shellt. T'0o such phylogenetic 
hypotheses the mathematical or dynamical study of the forms of 
shells lends no valid support. It we have two shells in which the 
constant angle of the spire be respectively 80° and 60°, that fact 
in itself does not at all justify an assertion that the one is more 
primitive, more ancient, or more “ancestral” than the other. 
Nor, if we find a third in which the angle happens to be 70°, 
does that fact entitle us to say that this shell is intermediate 
between the other two, in time, or in blood relationship, or in 
any other sense whatsoever save only the strictly formal and 
mathematical one. For it is evident that, though these particular 
arithmetical constants manifest themselves in visible and recog- 
nisable differences of form, yet they are not necessarily more 
deep-seated or significant than are those which manifest them- 
selves only in difference of magnitude; and the student of 
phylogeny scarcely ventures to draw conclusions as to the relative 
antiquity of two allied organisms on the ground that one happens 
to be bigger or less, or longer or shorter, than the other. 


* English edition, p. 537, 1900. The chapter is revised by Prof. Alpheus 
Hyatt, to whom the nomenclature is largely due. For a more copious terminology, 
see Hyatt, Phylogeny of an Acquired Characteristic, p. 422 seq., 1894. 

+ This latter conclusion is adopted by Willey, Zoological Results, p. 747, 1902. 


x1] OF VARIOUS CEPHALOPODS 549 


At the same time, while it is obviously unsafe to rest conclusions 
upon such features as these, unless they be strongly supported 
and corroborated in other ways,—for the simple reason that there 
is unlimited room for coincidence, or separate and independent 
attainment of this or that magnitude or numerical ratio,—yet on 
the other hand it is certain that, in particular cases, the evolution 
of a race has actually involved gradual increase or decrease in 
some one or more numerical factors, magnitude itself included,— 
that is to say increase or decrease in some one or more of the 
actual and relative velocities of growth. When we do meet with 
a clear and unmistakable series of such progressive magnitudes or 
ratios, manifesting themselves in a progressive series of “allied” 
forms, then we have the phenomenon of “orthogenesis.” For 
orthogenesis is simply that phenomenon of continuous lines or 
series of form (and also of functional or physiological capacity), 
which was the foundation of the Theory of Evolution, alike to 
Lamarck and to Darwin and Wallace; and which we see to exist 
whatever be our ideas of the “origin of species,” or of the nature 
and origin of “functional adaptations.” And to my mind, the 
mathematical (as distinguished from the purely physical) study 
of morphology bids fair to help us to recognise this phenomenon 
of orthogenesis in many cases where it is not at once patent to 
the eye; and also, on the other hand, to warn us, in many other 
cases, that even strong and apparently complex resemblances in 
form may be capable of arising independently, and may sometimes 
signify no more than the equally accidental numerical coincidences 
which are manifested in identity of length or weight, or any other 
simple magnitudes. 


I have already referred to the fact that, while in general a 
very great and remarkable regularity of form is characteristic of 
the molluscan shell, that complete regularity is apt to be departed 
from. We have clear cases of such a departure in Pupa, Clausilia, 
and various Bulimi, where the enveloping cone of the spire is 
not a right cone but a cone whose sides are curved. It is plain 
that this condition may arise in two ways: either by a gradual 
change in the ratio of growth of the whorls, that is to say in 
the logarithmic spiral itself, or by a change in the velocity of 


550 THE LOGARITHMIC SPIRAL [CH. 


translation along the axis, that is to say in the helicoid which, 
in all turbinate shells, is superposed upon the spiral. Very careful 
measurements will be necessary to determine to which of these 
factors, or in what proportions to each, the phenomenon is due. 
But in many Ammonitoidea where the helicoid factor does not 
enter into the case, we have a clear illustration of gradual and 
marked changes in the spiral angle itself, that is to say of the ratio 
of growth corresponding to increase of vectorial angle. We have 
seen from some of Naumann’s and Grabau’s measurements that 
such a tendency to vary, such an acceleration or retardation, 
may be detected even in Ammonites which present nothing 
abnormal to the eye. But let us suppose that the spiral angle 
increases somewhat rapidly; we shall then get a spiral with 
gradually narrowing whorls, and this condition is characteristic 


(pl ui 


Soe AL Wy, 


2. 


ez 
<== 
SSS 


Fig. 281. An ammonitoid shell (AT dontea sien to shew change of 
curvature. 


Tl in 


of Oekotraustes, a subgenus of Ammonites. If on the other hand, 
the angle a gradually diminishes, and even falls away to zero, we 
shall have the spiral curve opening out, as it does in Scaphites, 
Ancyloceras and Lituites, until the spiral coil is replaced by a spiral 
curve so gentle as to seem all but straight. Lastly, there are a 
few cases, such as Bellerophon expansus and some Goniatites, 
where the outer spiral does not perceptibly change, but the whorls 
become more “embracing” or the whole shell more involute. 
Here it is the angle of retardation, the ratio of growth between 
the outer and inner parts of the whorl, which undergoes a gradual 
change. 


In order to understand the relation of a close-coiled shell to 
one of its straighter congeners, to compare (for example) an 


x1] OF VARIOUS CEPHALOPODS 551 


Ammonite with an Orthoceras, it is necessary to estimate the 
length of the right cone which has, so to speak, been coiled up 
into the spiral shell. Our problem then is, To find the length of 
a plane logarithmic spiral, in terms of the radius and the constant 
angle a. In the annexed diagram, if OP be a radius vector, OQ 
a line of reference perpendicular to OP, and P@ a tangent to the 
curve, PQ, or seca, is equal in length to the spiral arc OP. And 
this is practically obvious: for PP’/PR’ = ds/dr = seca, and 
therefore sec a = s/r, or the ratio of arc to radius vector. 
Accordingly, the ratio of /, the total length, to r, the radius 


Fig. 282. 


vector up to which the total length is to be measured, is expressed 
by a simple table of secants; as follows: 


a Ur a Ur 

5° 1-004 87° 19-1 
10 1-015 88 28-7 
20 (1-064 89 57°3 
30 1-165 89° 10° 68°8 
40 1-305 20 85:9 
50 1-56 30 114-6 
60 2-0 40 171-9 
70 2-9 50 343°8 
75 3°9 55 687:5 
80 58 59 3437°7 
85 11-5 90 Infinite 
86 14:3 


Putting the same table inversely, so as to shew the total 


552 THE LOGARITHMIC SPIRAL (cH. 


length in whole numbers, in terms of the radius, we have as 


follows : 
Total length (in terms 


of the radius) Constant angle 
2 60° 
3 7 iN acs he 
4 75 32 
5 78 28 
10 84 16 
20 87 8 
30 88 6 
40 88 34 
50 88 51 
100 89 26 
1000 89 56’ 36” 
10,000 89 59 30 


Accordingly, we see that (1), when the constant angle of the 
spiral is small, the spiral itself is scarcely distinguishable from 
a straight line, and its length is but very little greater than that 
of its own radius vector. This remains pretty much the case for 
a considerable increase of angle, say from 0° to 20° or more; 
(2) for a very considerably greater increase of the constant angle, 
say to 50° or more, the shell would only have the appearance of 
a gentle curve; (3) the characteristic close coils of the Nautilus 
or Ammonite would be typically represented only when the 
- constant angle lies within a few degrees on either side of about 
80°. The coiled up spiral of a Nautilus, with a constant angle 
of about 80°, is about six times the length of its radius vector, 
or rather more than three times its own diameter: while that of 
an Ammonite, with a constant angle of, say, from 85° to 88°, is 
from about six to fifteen times as long as its own diameter. And 
(4) as we approach an angle of 90° (at which point the spiral 
vanishes in a circle), the length of the coil increases with enormous 
rapidity. Our spiral would soon assume the appearance of the 
close coils of a Nummulite, and the successive increments of 
breadth in the successive whorls would become inappreciable to 
the eye. The logarithmic spiral of high constant angle would. 
as we have already seen, tend to become indistinguishable, without 
the most careful measurement, from an Archimedean spiral. 
And it is obvious, moreover, that our ordinary methods of 


XI] OF VARIOUS CEPHALOPODS 553 


determining the constant angle of the spiral would not in these 
cases be accurate enough to enable us to measure the length of 
the coil: we should have to devise a new method, based on the 
measurement of radii or diameters over a large number of whorls. 

The geometrical form of the shell involves many other beautiful 
properties, of great interest to the mathematician, but which it 
is not possible to reduce to such simple expressions as we have 
been content to use. For instance, we may obtain an equation 
which shall express completely the surface of any shell, in terms 
of polar or of rectangular coordinates (as has been done by Moseley 
and by Blake), or in Hamiltonian vector notation. It is likewise 
possible (though of little interest to the naturalist) to determine 
the area of a conchoidal surface, or the volume of a conchoidal 
solid, and to find the centre of gravity of either surface or solid*. 
And Blake has further shewn, with considerable elaboration, how 
we may deal with the symmetrical distortion, due to pressure, 
which fossil shells are often found to have undergone, and how 
we may reconstitute by calculation their original undistorted 
form,—a problem which, were the available methods only a little 
easier, would be very helpful to the palaeontologist; for, as 
Blake himself has shewn, it is easy to mistake a symmetrically 
distorted specimen of (for instance) an Ammonite, for a new and 
distinct species of the same genus. But it is evident that to deal 
fully with the mathematical problems contained in, or suggested 
by, the spiral shell, would require a whole treatise, rather than 
a single chapter of this elementary book. Let us then, leaving 
mathematics aside, attempt to summarise, and perhaps to extend, 
what has been said about the general possibilities of form in this 
class of organisms. 


% 
The Uniwalve Shell: a summary. 

The surface of any shell, whether discoid or turbinate, may be 
imagined to be generated by the revolution about a fixed axis of 
a closed curve, which, remaining always geometrically similar to 
itself, increases continually its dimensions: and, since the rate of 
growth of the generating curve and its velocity of rotation follow 
the same law, the curve traced in space by corresponding points 


* See Moseley, op. cit. pp. 361 seq. 


554 THE LOGARITHMIC SPIRAL [cH. 


in the generating curve is, in all cases, a logarithmic spiral. In 
discoid shells, the generating figure revolves in a plane perpendicular 
to the axis, as in Nautilus, the Argonaut and the Ammonite. 
In turbinate shells, it slides continually along the axis of revolu- 
tion, and the curve in space generated by any given point partakes, 
therefore, of the character of a helix, as well as of a logarithmic 
spiral; it may be strictly entitled a helico-spiral. Such turbinate 
or helico-spiral shells include the snail, the periwinkle and all the 
common typical Gastropods. 

The generating figure, as represented by the mouth of the 
shell, is sometimes a plane curve, of simple form; in other and 
more numerous cases, it becomes more complicated in form and 
its boundaries do not lie in one plane: but in such cases as these 
we may replace it by its “trace,” on a 
plane at some definite angle to the direction 
of growth, for instance by its form as it 
appears in a section through the axis of 
the helicoid shell. The generating curve 
is of very various shapes. It is circular 
in Scalaria or Cyclostoma, and in Spirula ; 
it may be considered as a segment of a 
circle in Natica or in Planorbis. It is 
approximately triangular in Conus, and 
rhomboidal in Solarium or Potamides. It 
is very commonly more or less elliptical: 
the long axis of the ellipse being parallel 
to the axis of theshell in Oliva and Cypraea ; 
all but perpendicular to it in many Trochi; 
and oblique to it in many well-marked 
cases, such as Stomatella, Lamellana, © 
Fig. 283. Section ofa spiral, Sagaretus haliotoides (Fig. 284) and Haliotis. 


or turbinate, univalve, 


ries ag In Nautilus pompilius it is approximately 
rion corrugaus, am. 5 ° . “7° 
(From Woodward.) . #Semi-ellipse, and in N. umbilicatus rather 


more than a semi-ellipse, the long axis 
lying in both cases perpendicular to the axis of the shell*. Its 


* In Nautilus, the “hood”? has somewhat different dimensions in the two 
sexes, and these differences are impressed upon the shell, that is to say upon its 
“generating curve.” The latter constitutes a somewhat broader ellipse in the 


x1] OF VARIOUS UNIVALVES 555 


form is seldom open to easy mathematical expression, save when 
it is an actual circle or ellipse; but an exception to this rule may 
be found in certain Ammonites, forming the group “Cordati,” 
where (as Blake points out) the curve is very nearly represented 
by a cardioid, whose equation is 7 = a (1 + cos 8). 

The generating curve may grow slowly or quickly; its growth- 
factor is very slow in Dentalium or Turritella, very rapid in Nerita, 
or Pileopsis, or Haliotis or the Limpet. It may contain the axis 
in its plane, as in Nautilus; it may be parallel to the axis, as in 
the majority of Gastropods; or it may be inclined to the axis, as 
it is Im a very marked degree in Haliotis. In fact, in Haliotis 
the generating curve is so oblique to the axis of the shell that 
the latter appears to grow by additions to one margin only (cf. 
Fig. 258), as in the case of the opercula of Turbo and Nerita 
referred to on p. 522; and this is what Moseley supposed it to do. 


Fig. 284. A, Lamellaria perspicua; B, Sigaretus halotoides. 
(After Woodward.) 


The general appearance of the entire shell is determined (apart 
from the form of its generating curve) by the magnitude of three 
angles; and these in turn are determined, as has been sufficiently 
explained, by the ratios of certain velocities of growth. These 
angles are (1) the constant angle of the logarithmic spiral (a); 
(2) in turbinate shells, the enveloping angle of the cone, or (taking 
half that angle) the angle (9) which a tangent to the whorls makes 
with the axis of the shell; and (3) an angle called the “angle of 
retardation”? (8), which expresses the retardation in growth of 
male than in the female. But this difference is not to be detected in the young; 
in other words, the form of the generating curve perceptibly alters with advancing 
age. Somewhat similar differences in the shells of Ammonites were long ago 


suspected, by D’Orbigny, to be due to sexual differences. (Cf. Willey, Natural 
Science, v1, p. 411, 1895; Zoological Results, p. 742, 1902.) 


556 THE LOGARITHMIC SPIRAL [CH. 


the inner as compared with the outer part of each whorl, and 
therefore measures the extent to which one whorl overlaps, or the 
extent to which it is separated from, another. 

The spiral angle (a) is very small in a impet, where it is usually 
taken as= 0°; but it is evidently of a significant amount, though 
obscured by the shortness of the tubular shell. In Dentalium 
it is still small, but sufficient to give the appearance of a regular 
curve; it amounts here probably to about 30° to 40°. In Haliotis 
it is from about 70° to 75°; in Nautilus about 80°; and it lies 
between 80° and 85°, or even more, in the majority of Gastropods. 

The case of Fissurella is curious. Here we have, apparently, 
a conical shell with no trace of spiral curvature, or (in other 
words) witha spiral angle which approximates to 0°; but in the 
minute embryonic shell (as in that of the limpet) a spiral convolution 
is distinctly to be seen. It would seem, then, that what we have 
to do with here is an unusually large growth-factor in the generating 
curve, which causes the shell to dilate into a cone of very wide 
angle, the apical portion of which has become lost or absorbed, 
and the remaining part of which is too short to show clearly its 
intrinsic curvature. In the closely allied Emarginula, there is 
likewise a well-marked spiral in the embryo, which however is 
still manifested in the curvature of the adult, nearly conical, shell. 
In both cases we have to do with a very wide-angled cone, and 
with a high retardation-factor for its inner, or posterior, border. 
The series is continued, from the apparently simple cone to the 
complete spiral, through such forms as Calyptraea. 

The angle a, as we have seen, is not always, nor rigorously, 
a constant angle. In some Ammonites it may increase with age, 
the whorls becoming closer and closer; in others it may decrease 
rapidly, and even fall to zero, the coiled shell then straightening 
out, as in Lituites and similar forms. It diminishes somewhat, 
also, in many Orthocerata, which are slightly curved in youth, 
but straight in age. It tends to increase notably in some common 
land-shells, the Pupae and Bulimi; and it decreases in Succinea. 

Directly related to the angle a is the ratio which subsists 
between the breadths of successive whorls. The following table 
gives a few illustrations of this ratio in particular cases, in addition © 
to those which we have already studied. 


XI] _OF VARIOUS UNIVALVES 55 
Ratio of breadth of consecutive whorls. 
Pointed Turbinates - Obtuse Turbinates and Discoids 
Telescopium fuscum oP 1-14 Conus virgo fe sae 1-25 
Acus subulatus ... ets 1-16 Conus litteratus ... ee 1-40 
*Turritella terebellata oe 1-18 Conus betulina ... Bhs 1-43 
*Turritella imbricata vas 1-20 *Helix nemoralis ... a 1-50 
Cerithium palustre nan 1-22 *Solarium perspectivum ... 1-50 
Turritella duplicata de 1-23 Solarium trochleare fos 1-62 
Melanopsis terebralis ... 1-23 Solarium magnificum  ... 1-75 
Cerithium nodulosum ... 1-24 *Natica aperta ... ab 2-00 
*Turritella carinata =o 1-25 Euomphalus pentangulatus 2-00 
Acus crenulatus ... ase 1-25 Planorbis corneus ba 2-00 
Terebra maculata (Fig. 285) 1-25 Solaropsis pellis-serpentis 2-00 
*Cerithium ligritarum ... 1-26 Dolium zonatum ... ie 2-10 
Acus dimidiatus ... nee 1-28 *Natica glaucina ... aoe 3-00 
Cerithium sulcatum iss 1-32 Nautilus pompilius ot 3-00 
Fusus longissimus gas 1-34 Haliotis excavatus aie 4-20 
*Pleurotomaria conoidea ... 1-34 Haliotis parvus ... ues 6-00 
Trochus niloticus (Fig. 286) 1-41 Delphinula atrata ae 6-00 
Mitra episcopalis ... =r 1-43 Haliotis rugoso-plicata ... 9-30 
Fusus antiquus ... di 1-50 Haliotis viridis... 3 10:00 
Scalaria pretiosa ... a 1-56 

Fusus colosseus ... at 1-71 


Phasianella bulloides ae 1-80 
Helicostyla polychroa Sec 2-00 


Those marked * from Naumann; the rest from Macalister ft. 


In the case of turbinate shells, we require to take into account 
the angle @, in order to determine the spiral angle a from the 
ratio of the breadths of consecutive whorls; for the short table 
given on p. 534 1s only applicable to discoid shells, in which 
the angle @ is an, angle of 90°. Our formula, as mentioned on 


p. 518 now becomes 
nS e27 sind cot a 


For this formula I have worked out the following table. 


+ Macalister, Alex., Observations on the Mode of Growth of Discoid and 
Turbinated Shells, P. R. S. xvui, pp. 529-532, 1870. 


oF €¢ 
eeetet 
Té +9 
&¢ 69 
8€ GL 
Ge OL 
PE LL 
€PF 8L 
g 08 
cy I8 
Gh &8 
8I 98 
8g L8 
8 68 

006 


0Z &9 
cr LG 
OL +9 
ge 69 
Go GL 
0G 9L 
G6 LL 
6& 8L 
99 6L 
gé 18 
LE &8 
ST 98 
9¢ L8 
/L 68 
008 


0g 
iV 


L8 
o6 8 


o0L 


‘9 abun-vuas wordy ayy fo sanjoa snorwma sof ‘yays ay fo sjsoym aarssaoons 


S7.6r 
SI 9 
OL 19 
v 19 
[€ €L 
& FL 
Gy GL 
G LL 
Ge 8L 
96 08 
GP G8 
vy 98 
6€ L8 
M 68 

009 


eo 0 
ine) 


LI 
61 
6G 
éL 
1G 


LY 
6g 
8¢ 
v9 
IL 
GL 
PL 
GL 
LL 
6L 
18 
g8 
L8 


GS 088 
009 


og 


If 
GP 
€¢ 
09 
89 
69 
IL 
GL 
FL 
LL 
08 
v8 
98 


/O€ 088 
oOF 


0G 
cP 
GG 
cP 
gg 
GG 
Or 
ST 
cP 
GSP 
VE 
66 


9g 


PE 
8E 
oF 
€¢ 
69 
v9 
99 
89 
OL 
€L 
LL 
68 


G8 


/9T 88 
06 


er 


0 &9 
Oey 
IIT €4 
Ié@ 6L 
G8 
/86 oL8 
006 


fo yypvasg fo sons uanjsao 07 burpuodsassooa 0 ajbun qosrds ayy fo sanjoa buamays a/qv 


CH. XI] OF VARIOUS UNIVALVES 559 


From this table, by interpolation, we may easily fill in the 
approximate values of a, as soon as we have determined the 
apical angle @ and measured the ratio R; as follows: 


Turriiella sp. 

Cerithium nodulosum 
Conus virgo 5 
Mitra episcopalis ... 
Scalaria pretiosa 
Phasianella bulloides 
Solarium perspectivum 
Natica aperta 

Planorbis corneus . ; 
Huomphalus pentangulatus 


R 
1-12 
1-24 
1-25 
1-43 
1-56 
1-80 
1-50 
2-00 
2-00 
2-00 


0 a 

ae 81° 
15 82 
70 88 
16 78 
26 81 
26 80 
53 85 
70 83 
£O 84 
90 84 


We see from this that shells so different in appearance as 
Cerithium, Solarium, Natica and Planorbis differ very little indeed 
in the magnitude of the spiral angle a, that is to say in the relative 


velocities of radial and tangential growth. 


that the difference in their form 
mainly depends: that is to say the 
amount of longitudinal shearing, 
or displacement parallel to the axis 
of the shell. 

The enveloping angle, or rather 
semi-angle (@), of the cone may be 
taken as 90° in the discoid shells, 
such as Nautilus and Planorbis. It 
is still a large angle, of 70° or 75°, 
in Conus or in Cymba, somewhat 
less in Cassis, Harpa, Dolium or 
Natica : iis about. 50°" to. 55 IM. 
the various species of Solarium, 
about 35° in the typical Trochi, 
such as 7. niloticus.or T. zizyphinus, 
and about 25° or 26° in Scalaria 
pretiosa and Phasianella bulloides ; it 
becomes a very acute angle, of 
15°, 10°, or even less, in Kulima, 


It is upon the angle 6 


Terebra maculata, L. 


Turritella or Cerithium. The costly Conus gloria-maris, one of the 


560 THE LOGARITHMIC SPIRAL [CH. 


great treasures of the conchologist, differs from its congeners in 
no important particular save in the somewhat “produced” spire, 
that is to say in the comparatively low value of the angle 0. 

A variation with advancing age of # is common, but (as Blake 
points. out) it is often not to be distinguished or disentangled from 
an alteration of a. Whether alone, or combined with a change in. 
a, we find it in all those many Gastropods whose whorls cannot 
all be touched by the same enveloping cone, and whose spire is 
accordingly described as concave or convex. The former condition, 
as we have it in Cerithium, and in the cusp-like spire of Cassis, 


Fig. 286. Trochus niloticus, L. 


Dolium and some Cones, is much the commoner cf the two. 
And such tendency to decrease may lead to @ becoming a negative 
angle; in which case we have a depressed spire, as in the 
Cypraeae. 

When we find a “reversed shell,” a whelk or a snail for instance 
whose spire winds to the left instead of to the right, we may 
describe it mathematically by the simple statement that the angle 
6 has changed sign. In the genus Ampullaria, or Apple-snails, 
inhabiting tropical or sub-tropical rivers, we have a remarkable 
condition; for in certain “species” the spiral turns to the right, 
in others to the left, and in others again we have a flattened 


x1] OF BIVALVE SHELLS 561 


“discoid” shell; and furthermore we have numerous intermediate 
stages, on either side, shewing right and left-handed spirals of 
varying degrees of acuteness*. In this case, the angle 6 may be 
said to vary, within the limits of a genus, from somewhere about 
35° to somewhere about 125°. 

The angle of retardation (f) is very small in Dentalium and 
Patella; it is very large in Hahotis. It becomes infinite in 
Argonauta and in Cypraea. Connected with the angle of retarda- 
tion are the various possibilities of contact or separation, in various 
degrees, between adjacent whorls in the discoid, and between 
both adjacent and opposite whorls in the turbinated shell. But 
with these phenomena we have already dealt sufficiently. 


Of Bivalve Shells. 


Hitherto we have dealt only with univalve shells, and it is in 
these that all the mathematical problems connected with the 
spiral, or helico-spiral, are best illustrated. But the case of the 
bivalve shell, of Lamellibranchs or of Brachiopods, presents no 
essential difference, save only that we have here to do with two 
conjugate spirals, whose two axes have a definite relation to one 
another, and some freedom of rotatory movement relatively to 
one another. 

The generating curve is particularly well seen in the bivalve, 
where it simply constitutes what we call “the outline of the shell.” 
It is for the most part a plane curve, but not always; for there 
are forms, such as Hippopus, Tridacna and many Cockles, or 
Rhynchonella and Spirifer among the Brachiopods, in which the 
edges of the two valves interlock, and others, such as Pholas, 
Mya, etc., where in part they fail to meet. In such cases as these 
the generating curves are conjugate, having a similar relation, but 
of opposite sign, to a median plane of reference. A great variety 
of form is exhibited by these generating curves among the bivalves. 
In a good many cases the curve is approximately circular, as in 
Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular in 
Argiope. It is approximately elliptical in Orthis and in Anodon; 
it may be called semi-elliptical in Spirifer. It is a nearly rectilinear 

* See figures in Arnold Lang’s Comparative Anatomy (English translation), 1, 
p. 161, 1902. 


7. G. 36 


562 THE LOGARITHMIC SPIRAL 0 eae 


triangle in Lithocardium, and a curvilinear triangle in Mactra. 
Many apparently diverse but more or less related forms may be 
shewn to be deformations of a common type, by a simple applica- 
tion of the mathematical theory of “Transformations,” which we 
shall have to study in a later chapter. In such a series as is 
furnished, for instance, by Gervillea, Perna, Avicula, Modiola, 
Mytilus, etc., a “simple shear” accounts for most, 1f not all, of 
the apparent differences. 

Upon the surface of the bivalve shell we usually see with great 
clearness the “lines of growth” which represent the successive 
margins of the shell, or in other words the successive positions 
assumed during growth by the growing generating curve; and 
we have a good illustration, accordingly, of how it is characteristic 
of the generating curve that it should constantly increase, while 
never altering its geometric similarity. 

Underlying these “lines of growth,” which are so characteristic 
of a molluscan shell (and of not a few other organic formations), 
there is, then, a “law of growth” which we may attempt to enquire 
into and which may be illustrated in various ways. The simplest 
cases are those in which we can study the lines of growth on a 
more or less flattened shell, such as the one valve of an oyster, 
a Pecten or a Tellina, or some such bivalve mollusc. Here around 
an origin, the so-called “umbo” of the shell, we have a series of 
curves, sometimes nearly circular, sometimes elliptical, and often 
asymmetrical; and such curves are obviously not “concentric,” 
though we are often apt to call them so, but are always “co-axial.” 
This manner of arrangement may be illustrated by various 
analogies. We might for instance compare it to a series of waves, 
radiating outwards from a point, through a medium which offered 
a resistance increasing, with the angle of divergence, according to 
some simple law. We may find another, and perhaps a simpler 
illustration as follows: 

In a very simple and beautiful theorem, Galileo shewed that, 
if we imagine a number of inclined planes, or gutters, sloping 
downwards (in a vertical plane) at various angles from a common 
starting-point, and if we imagine a number of balls rolling each 
down its own gutter under the influence of gravity (and without 
hindrance from friction), then, at any given instant, the locus of 


XI] OF BIVALVE SHELLS 563 


all these moving bodies is a circle passing through the point of 
origin. For the acceleration along any one of the sloping paths, 
for instance AB (Fig. 287), is such 


A 
that 
AB = $9 cos 0. ¢ 
=p AAC. i, 
Therefore B 
= 2/9 ..AC. 

That is to say, all the balls 
reach the circumference of the g 
circle at the same moment as the 
ball which drops vertically from pce 

Fig. 287. 


A to C. 

Where, then, as often happens, the generating curve of the 
shell is approximately a circle passing through the point of origin, 
we may consider the acceleration of growth along various radiants 
to be governed by a simple mathematical law, closely akin to 
that simple law of acceleration which governs the movements of 
a falling body. And, mutatis mutandis, a similar definite law 
_ underlies the cases where the generating curve is continually 
elliptical, or where it assumes some more complex, but still regular 
and constant form. 

It is easy to extend the proposition to the particular case where 
the lines of growth may be considered elliptical. In such a case 
we have w/a? + y?/b? = 1, where a and 6 are the major and minor 
axes of the ellipse. 

Or, changing the origin to the vertex of the figure 


giving 


Then, transferring to polar coordinates, where 7.cos@=4@, 
7.sin@=y, we have 


r.cos?@ 2cos@ psi) 


Q, 
a a b 


36—2 


564 


THE LOGARITHMIC SPIRAL [CH. 


which is equivalent to 


re 2ab? cos 6 
b? cos? 6 + a? sin? @’ 


or, eliminating the sine-function, 


ae 2ab? cos 6 
~ (b? — a?) cos? 6 + a? 


Obviously, in the case when a = b, this gives us the circular 
system which we have already considered. For other values, or 
ratios, of a and 6, and for all values of 0, we can easily construct 
a table, of which the following is a sample: 


a/b=1/3 


1-0 
1-01 
1-05 
1-115 
1-21 
1-34 
1-50 
1-59 
1-235 
0-0 


Chords of an ellipse, whose major and minor axes (a, b) 


are in. certain given ratios. 


1/2 2/3 1/1 3/2 2/1 3/1 
1-0 1-0 1-0 1-0 1-0 10 
1-01 1-002 -985 948 -902 ‘793 
1:03 1-005 -940 -820 695 485 
1-065 1-005 -866 666 495 289 
Ll 995 -766 505 +342 178 
1-145 952 643 ‘372 -232 113 
1-142 -857 -500 -258 152 O71 
1-015 -670 +342 163 092 042 
635 ‘375 ‘174 -078 045 -020 
0-0 . 0-0 0-0 0-0 0-0 0-0 


The coaxial ellipses which we then draw, from the values given 
in the table, are such as are shewn in Fig. 288 for the ratio 


a/b = *, and in Fig. 289 for the ratio a/b = 3; 
these are fair approximations to the actual 
outlines, and to the actual arrangement of the 
hnes of growth, in such forms as Solecurtus or 
Cultellus, and in Tellina or Psammobia. It is 
not difficult to introduce a constant into our 
equation to meet the case of a shell which is 
somewhat unsymmetrical on either side of the 
median axis. It is a somewhat more trouble- 
some matter, however, to bring these con- 
figurations into relation with a “law of 
growth,” as was so easily done in the case 
of the circular figure: in other words, to 


x1] OF BIVALVE SHELLS 565 


formulate a law of acceleration according to which points starting 
from the origin O, and moving along radial lines, would all lie, at 
any future epoch, on an ellipse passing through O; and this 
calculation we need not enter into. 

All that we are immediately concerned with is the simple fact 
that where a velocity, such as our rate of growth, varies with its 
direction,—varies that is to say as a function of the angular 
divergence from a certain axis,—then, in a certain simple case, 
we get lines of growth laid down as a system of coaxial circles, 
and, when the function is a more complex one, as a system of 
ellipses or of other more complicated coaxial figures, which figures 
may or may not be symmetrical on either side of the axis. Among 


0. 


90° 
—— 
SES A 
NN Ast wes = 80: 
BSCS Ne cana 
Ss \ ~ —— 
oS ; =<. 
ae 
aoe . 
ie NM 50° 
\. \ x 40° 
O20: 
eee eo 
Fig. 289. 


our bivalve mollusca we shall find the lines of growth to be 
approximately circular in, for instance, Anomia; in Lima (e.g. 
L. subauriculata) we have a system of nearly symmetrical ellipses 
with the vertical axis about twice the transverse; in Solen pellu- 
cidus, we have again a system of lines of growth which are not far 
from being symmetrical ellipses, in which however the transverse 
is between three and four times as great as the vertical axis. In 
the great majority of cases, we have a similar phenomenon with 
the further complication of slight, but occasionally very consider- 
able, lateral asymmetry. 

In certain little Crustacea (of the genus Estheria) the carapace 
takes the form of a bivalve shell, closely simulating that of a 


566 THE LOGARITHMIC SPIRAL [CH. 


lamellibranchiate mollusc, and bearing lines of growth in all 
respects analogous to or even identical with those of the latter. 
The explanation is very curious and interesting. In ordinary 
Crustacea the carapace, like the rest of the chitinised and calcified 
integument, is shed off in successive moults, and is restored again 
asa whole. But in Estheria (and one or two other small crustacea) 
the moult is incomplete: the old carapace is retained, and the 
new, growing up underneath it, adheres to it like a liming, and 
projects beyond its edge: so that in course of time the margins 
of successive old carapaces appear as “lines of growth” upon the 
surface of the shell. In this mode of formation, then (but not 
in the usual one), we obtain a structure which “is partly old and 
partly new,” and whose successive increments are all similar, 
similarly situated, and enlarged in a continued progression. We 
have, in short, all the conditions appropriate and necessary for 
the development of a logarithmic spiral; and this logarithmic 
spiral (though it is one of small angle) gives its own character to 
the structure, and causes the little carapace to partake of the 
characteristic conformation of the molluscan shell. 

The essential simplicity, as well as the great regularity of the 
“curves of growth” which result in the familiar configurations of 
our bivalve shells, sufficiently explain, in a general way, the ease 
with which they may be imitated, as for instance in the so-called 
“artificial shells’? which Kappers has produced from the conchoidal 
form and lamination of lumps of melted and quickly cooled 
paraffin *. 


In the above account of the mathematical form of the bivalve shell, we 
have supposed, for simplicity’s sake, that the pole or origin of the system is 
at a point where all the successive curves touch one another. But such an 
arrangement is neither theoretically probable, nor is it actually the case; 
for it would mean that in a certain direction growth fell, not merely to a 
minimum, but to zero. As a matter of fact, the centre of the system (the 
“umbo” of the conchologists) lies not at the edge of the system, but very 
near to it; in other words, there is a certain amount of growth all round. 
But to take account of this condition would involve more troublesome mathe- 
matics, and it is obvious that the foregoing illustrations are a sufficiently near 
approximation to the actual case. 


* Kappers, C. U. A., Die Bildung kiinstlicher Molluskenschalen, Zeitschr. f. 
allg. Physiol. va, p. 166, 1908. 


xi] ; OF BIVALVE SHELLS 567 


Among the bivalves the spiral angle (a) is very small in the 
flattened shells, such as Orthis, Lingula or Anomia. It is larger, 
as a rule, in the Lamellibranchs than in the Brachiopods, but in 

the latter it is of considerable magnitude among the Pentameri. 
Among the Lamellibranchs it is largest in such forms as Isocardia 
and Diceras, and in the very curious genus Caprinella; in all of 
these last-named genera its magnitude leads to the production of 
a spiral shell of several whorls, precisely as in the univalves. The 
angle is usually equal, but of opposite sign, in the two valves of 
the Lamellibranch, and usually of opposite sign but unequal in 


ig. 290. Caprinella adversa. Fig. 291. Section of Productus 
(After Woodward.) (Strophomena) sp. (From 
Woods.) 


the two valves of the Brachiopod. It is very unequal in many 
Ostreidae, and especially in such forms as Gryphaea, or in Capri- 
nella, which is a kind of exaggerated Gryphaea. Occasionally it 
is of the same sign in both valves (that is to say, both valves curve 
the same way) as we see sometimes in Anomia, and much better 
in Productus or Strophomena. 

Owing to the large growth-factor of the generating curve, and 
the comparatively small angle of the spiral, the whole shell seldom 
assumes a spiral form so conspicuous as to manifest in a typical 
way the helical twist or shear which is so conspicuous in the 


568 THE LOGARITHMIC SPIRAL | [CH. 


majority of univalves, or to let us measure or estimate the 
magnitude of the apical angle (@) of the enveloping cone. This 
however we can do in forms like Isocardia and Diceras; while in 
Caprinella we see that the whorls lie in a plane perpendicular to 
the axis, forming a discoidal spire. As in the latter shell, so also 
universally among the Brachiopods, there is no lateral asymmetry 
in the plane of the generating curve such as to lead to the develop- 
ment of a helix; but in the majority of the Lamellibranchiata 
it is obvious, from the obliquity of the lines of growth, that the 
angle @ is significant in amount. 


The so-called “spiral arms” of Spirifer and many other 
Brachiopods are not difficult to explain. They begin as a single 
structure, in the form of a loop of 
shelly substance, attached to the 
dorsal valve of the shell, in the 
neighbourhood of the hinge. This 
loop has a curvature of its own, similar 
to but not necessarily identical with 
that of. the valve to which it is 
attached; and this curvature will tend 
to be developed, by continuous and 
symmetrical growth, into a_ fully 

; ae ae formed logarithmic spiral, so far as 
Big. 292: Skeletal loop of . - : 
Terebratula. (From Woods.) it 18 permitted to do so under the 
constraint of the shell in which it is 
contained. In various Terebratulae we see the spiral growth of 
the loop,-more or less flattened and distorted by the restraining 
pressure of the ventral valve. In a number of cases the loop 
remains small, but gives off two nearly parallel branches or off- 
shoots, which continue to grow. And these, starting with just 
such a shght curvature as the loop itself possessed, grow on and 
on till they may form close-wound spirals, always provided that 
the “spiral angle” of the curve is such that the resulting spire 
can be freely contained within the cavity of the shell. Owing to. 
the bilateral symmetry of the whole system, the case will be rare, 
and unlikely to occur, in which each separate arm will coil strictly 
mm a plane, so as to constitute a discoid spiral; for the original 


XI] OF BIVALVE SHELLS 569 


direction of each of the two branches, parallel to the valve (or 
nearly so) and outwards from the middle line, will tend to con- 
stitute a curve of double curvature, and so, on further growth, 
to develop into a helicoid. This is what actually occurs, in the 
great majority of cases. But the curvature may be such that 
the helicoid grows outwards from the middle line, or inwards 
towards the middle line, a very slight difference in the initial 
curvature being sufficient to direct the spire the one way or the 
other; the middle course of an undeviating discoid spire will be 
rare, from the usual lack of any obvious controlling force to prevent 
its deviation. The cases in which the helicoid spires point towards, 
or point away from, the middle line are ascribed, in zoological 
classification, to particular “families” of Brachiopods, the former 


Fig. 293. Spiral arms of Fig. 294. Inwardly directed 
Spirifer. (From Woods.) spiral arms of Atrypa. 


condition defining (or helping to define) the Atrypidae and the 
latter the Spiriferidae and Athyridae. It is obvious that the 
incipient curvature of the arms, and consequently the form and 
direction of the spirals, will be influenced by the surrounding 
pressures, and these in turn by the general shape of the shell. 
We shall expect, accordingly, to find the long outwardly directed 
spirals associated with shells which are transversely elongated, as 
Spirifer is; while the more rounded Atrypas will tend to the 
opposite condition. In a few cases, as in Cyrtina or Reticularia, 
where the shell is comparatively narrow but long, and where the 
uncoiled basal support of the arms is long also, the spiral coils 
into which the latter grow are turned backwards, in the direction 
where there is room for them. And in the few cases where the 
shell is very considerably flattened, the spirals (if they find room 


_ 570 THE LOGARITHMIC SPIRAL [CH. 


to grow at all) will be constrained to do so in a discoid or nearly 
discoid fashion, and this is actually the case in such flattened 
forms as Koninckina or Thecidium. 


The Shells of Pteropods. 


While mathematically speaking we are entitled to look upon 
the bivalve shell of the Lamellibranch as consisting of two distinct 
elements, each comparable to the entire shell of the univalve, we 
have no biological grounds for such a statement; for the shell 
arises from a single embryonic origin, and afterwards becomes split 
into portions which constitute the two separate valves. We can 
perhaps throw some indirect light upon this phenomenon, and 
upon several other phenomena connected with shell-growth, by 
a consideration of the simple conical or tubular shells of the 
Pteropods. The shells of the latter are in few cases suitable for 
simple mathematical investigation, but nevertheless they are of 
very considerable interest in connection with our general problem. 

The morphology of the Pteropods is by no 
means well understood, and in speaking of 
them I will assume that there are still 


qaaat 


— grounds for believing (in spite of Boas’ 
—S and Pelseneer’s arguments) that they are 

oes directly related to, or may at least be 
te 


directly compared with, the Cephalopoda*. 

The simplest shells among the Pteropods 
have the form of a tube, more or less 
cylindrical (Cuvierina), more often conical 
(Creseis, Clio); and this tubular shell (as 
we have already had occasion to remark, 
on p. 258), frequently tends, when it is 
very small and delicate, to assume the 
Fis. 295. Pteropod shells; Character of an unduloid. (In such a case 

(1) Cuxterina columnella; it 1s more than likely that the tiny shell, 

(2) Cleodora chierchiaes 4» that portion of it which constitutes the 


(3) C. pygmaea. (After _ i 
Boas.) unduloid, has not grown by successive 


* We need not assume a close relationship, nor indeed any more than such a 
one as permits us to compare the shell of a Nautilus with that of a Gastropod. 


xt] THE SHELLS OF PTEROPODS — 571 


increments or “rings of growth,” but has developed as a whole.) 
A thickened “rib” is often, perhaps generally, present on the 
dorsal side of the little conical shell. In a few cases (Limacina, 


Fig. 296. Diagrammatic transverse sections, or outlines of the mouth, in certain 
Pteropod shells: A, B, Cleodora australis; C, C. pyramidalis; D, C. balantium ; 
E, C. cuspidata. (After Boas.) 


— 
v 


Fig. 297. Shells of thecosome Pteropods (after Boas). (1) Cleodora 


cuspidata; (2) Hyalaea trispinosa: (3) H. globulosa; (4) H. wneinata; 
(5) H. inflexa. 


Peraclis) the tube becomes spirally coiled, in a normal logarithmic 


spiral or helico-spiral. 
In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is 


curiously modified. In the first place, its cross-section, originally 


572 THE LOGARITHMIC SPIRAL [CH. 


circular or nearly so, becomes flattened or compressed dorso- 
ventrally ; and the angle, or rather edge, where dorsal and ventral 
walls meet, becomes more and more drawn out into a ridge or 
keel. Along the free margin, both of the dorsal and the ventral 
portion of the shell, growth proceeds with a regularly varying 
velocity, so that these margins, or lips, of the shell become regularly 
curved or markedly sinuous. At the same time, growth in a 
transverse direction proceeds with an acceleration which manifests 
itself in a curvature of the sides, replacing the straight borders of 
the original cone. In other words, the cross-section of the cone, 
or what we have been calling the generating curve, increases its 
dimensions more rapidly than its distance from the pole. 


Fig. 298. Cleodora cuspidata. 


In the above figures, for instance in that of Cleodora cuspidata, 
the markings of the shell which represent the successive edges of 
the lip at former stages of growth, furnish us at once with a 
“graph” of the varying velocities of growth as measured, radially, 
from the apex. We can reveal more clearly the nature of these 
variations in the following way which is simply tantamount to 
converting our radial into rectangular coordinates. Neglecting 
curvature (if any) of the sides and treating the shell (for simplicity’s 
sake) as a right cone, we lay off equal angles from the apex O, 
along the radii Oa, Ob, etc. If we then plot, as vertical equi- 
distant ordinates, the magnitudes Oa, Ob...OY, and again on to 
Oa’, we obtain a diagram such as the following (Fig. 299); by 


XI] THE SHELLS OF PTEROPODS 573 


help of which we not only see more clearly the way in which the 
growth-rate varies from point to point, but we also recognise 
much better than before, the similar nature of the law which 
governs this variation in the different species. 


aorGid er oN 


Habe 


O 


XxX x 
Fig. 299. Curves obtained by transforming radial ordinates, as in Fig. 298, into 
vertical equidistant ordinates. 1, Hyalaea trispinosa; 2, Cleodora cuspidata. 


Furthermore, the young shell having become differentiated into a 
dorsal and a ventral part, marked off from one another by a lateral 
edge or keel, and the inequality of growth being such as to cause 


Fig. 300. Development of the shell of Hyalaea (Cavolinia) tridentata, Forskal: 
the earlier stages being the “ Pleuropus longifilis”’ of Troschel. (After Tesch.) 


each portion to increase most rapidly in the median line, it follows 
that the entire shell will appear to have been split into a dorsal 
and a ventral plate, both connected with, and projecting from, 


574 THE LOGARITHMIC SPIRAL [CH. 


what remains of the original undivided cone. Putting the same 
thing in other words, we may say that the generating figure, which 
lay at first in a plane perpendicular to the axis of the cone, has 
now, by unequal growth, been sharply bent or folded, so as to 
lie approximately in two planes, parallel to the anterior and 
posterior faces of the cone. We have only to imagine the apical 
connecting portion to be further reduced, and finally to disappear 
or rupture, and we should have a bivalve shell developed out of 
the original simple cone. 

In its outer and growing portion, the shell of our Pteropod 
now consists of two parts which, though still connected together 
at the apex, may be treated as growing practically independently. 
The shell is no longer a simple tube, or simple cone, in which 
regular inequalities of growth will lead to the development of a 
spiral; and this for the simple reason that we have now two 
opposite maxima of growth, instead of a maximum on the one side 
and a minimum on the other side of our tubular shell. As a matter 
of fact, the dorsal and the ventral plate tend to curve in opposite 
directions, towards the middle line, the dorsal curving ventrally 
and the ventral curving towards the dorsal side. 

In the case of the Lamellibranch or the Brachiopod, it is quite 
possible for both valves to grow into more or less pronounced 
spirals, for the simple reason that they are hinged upon one another ; 
and each growing edge, instead of being brought to a standstill 
by the growth of its opposite neighbour, is free to move out of 
the way, by the rotation about the hinge of the plane in which 
it lies. 

But where, as in the Pteropod, there is no such hinge, the 
dorsal and ventral halves of the shell (or dorsal and ventral 
valves, if we may call them so), if they curved towards one 
another (as they do in a cockle), would soon interfere with 
one another’s progress, and the development of a pair of 
conjugate spirals would become impossible. Nevertheless, there 
is obviously, in both dorsal and ventral valve, a tendency to 
the development of a spiral curve, that of the ventral valve 
being more marked than that of the larger and overlapping 
dorsal one, exactly as in the two unequal valves of Terebratula. 
In many cases (e.g. Cleodora cuspidata), the dorsal valve or plate, 


xt] THE SHELLS OF PTEROPODS B75 


strengthened and stiffened by its midrib, is nearly straight, while 
the curvature of the other is well displayed. But the case will 
be materially altered and simplified if growth be arrested or 
retarded in either half of the shell. Suppose for instance that 
the dorsal valve grew so slowly that after a while, in comparison 
with the other, we might speak of it as being absent altogether: 
or suppose that it merely became so reduced in relative size as to 
form no impediment to the continued growth of the ventral one; 
the latter would continue to grow in the direction of its natural 
curvature, and would end by forming a complete and coiled 
logarithmic spiral. It would be precisely analogous to the spiral 
shell of Nautilus, and, in regard to its ventral position, concave 
towards the dorsal side, it would even deserve to be called directly 


Fig. 301. Pteropod shells, from the side: (1) Cleodora cuspidata; (2) Hyalaea 
longirostris; (3) H. trispinosa. (After Boas.) 


homologous with it. Suppose, on the other hand, that the ventral 
valve were to be greatly reduced, and even to disappear, the 
dorsal valve would then pursue its unopposed growth; and, were 
it to be markedly curved, it would come to form a logarithmic 
spiral, concave towards the ventral side, as is the case in the shell 
of Spirula*. Were the dorsal valve to be destitute of any marked 
curvature (or in other words, to have but a low spiral angle), it 
would form a simple plate, as in the shells of Sepia or Loligo. In- 
deed, in the shells of these latter, and especially in that of Sepia, 
we seem to recognise a manifest resemblance to the dorsal plate of 
the Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea; 

* Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely spiral or 


coiled over the back of the animal, not involute like Spirula”: Palaeontology, 
1861, p. 97; cf. Mem. on the Pearly Nautilus, 1832; also P.Z.S. 1878, p. 955. 


576 THE LOGARITHMIC SPIRAL [CH. 


the little “rostrum” of Sepia is but the apex of the primitive cone, 
and the rounded anterior extremity has grown according to a law 
precisely such as that which has produced the curved margin of 
the dorsal valve in the Pteropod.- The ventral portion of the ~ 
original cone is nearly, but not wholly, wanting. It is represented 
by the so-called posterior wall of the “siphuncular space.” In 
many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we 
still see at the posterior end of the “pen,” a vestige of the primitive 
cone, whose dorsal margin only has continued to grow; and the 
same phenomenon, on an exaggerated scale, is represented in the 
Belemnites. 

It is not at all impossible that we may explain on the same 
lines the development of the curious “operculum” of the Ammon- 
ites. This consists of a single horny plate (Anaptychus), or of 
a thicker, more calcified plate divided into two symmetrical 
halves (Aptychi), often found inside the terminal chamber of the 
Ammonite,-and occasionally to be seen lying in situ, as an 
operculum which partially closes the mouth of the shell; this 
structure is known to exist even in connection with the early 
embryonic shell. In form the Anaptychus, or the pair of con- 
jomed Aptychi, shew an upper and a lower border, the latter 
strongly convex, the former sometimes slightly concave, sometimes 
slightly convex, and usually shewing a median projection or 
slightly developed rostrum. From this “rostral” border the 
curves of growth start, and course round parallel to, finally 
constituting, the convex border. It is this convex border which 
fits into the free margin of the mouth of the Ammonite’s shell, 
while the other is applied to and overlaps the preceding whorl of 
the spire. Now this relationship is precisely what we should 
expect, were we to imagine as our starting-point a shell similar 
to that of Hyalaea, in which however the dorsal part of the split 
cone had become separate from the ventral half, had remained 
flat, and had grown comparatively slowly, while at the same time 
it kept slipping forward over the growing and coiling spire into 
which the ventral half of the original shell develops*. In short, 
I think there is reason to believe, or at least to suspect, that we 


* The case of Terebratula or of Gryphaea would be closely analogous, if the 
smaller valve were less closely connected and co-articulated with the larger. 


xd] THE SHELLS OF PTEROPODS 517 


have in the shell and Aptychus of the Ammonites, two portions 
of a once united structure; of which other Cephalopods retain 
not both parts but only one or other, one as the ventrally 
situated shell of Nautilus, the other as the dorsally placed shell 
for example of Sepia or of Spirula. 

In the case of the bivalve shells of the Lamellibranchs or of 
the Brachiopods, we have to deal with a phenomenon precisely 
analogous to the split and flattened cone of our Pteropods, save 
only that the primitive cone has been split into two portions, not 
incompletely as in the Pteropod (Hyalaea), but completely, so 
as to form two separate valves. Though somewhat greater 
freedom is given to growth now that the two valves are separate 
and hinged, yet still the two valves oppose and hamper one 
another, so that in the longitudinal direction each is capable of 
only a moderate curvature. This curvature, as we have seen, is 
recognisable as a logarithmic spiral, but only now and then does 
the growth of the spiral continue so far as to develop successive 
coils: as it does in a few symmetrical forms such as [socardia cor ; 
and as it does still more conspicuously in a few others, such as 
Gryphaea and Caprinella, where one of the two valves is stunted, 
and the growth of the other is (relatively speaking) unopposed. 


Of Septa. 

Before we leave the subject of the molluscan shell, we have 
still another problem to deal with, in regard to the form and 
arrangement of the septa which divide up the tubular shell into 
chambets, in the Nautilus, the Ammonite and their allies (Fig. 
304, etc.). 

The existence of septa in a Nautiloid shell may probably be 
accounted for as follows. We have seen that it is a property of 
a cone that, while growing by increments at one end only, it 
conserves its original shape: therefore the animal ,within, which 
(though growing by a different law) also conserves its shape, will 
continue to fill the shell if it actually fills it to begin with: as 
does a snail or other Gastropod. ‘But suppose that our mollusc 
fills a part only of a conical shell (as it does in the case of Nautilus) ; 
then, unless it alter its shape, it must move upward as it grows in 
the growing cone, until it come to occupy a space similar in form 

T G. 37 


578 THE LOGARITHMIC SPIRAL [CH. 


to that which it occupied before: just, indeed, as a little ball 
drops far down into the cone, but a big one must stay farther up. 
Then, when the animal after a period of growth has moved farther 
up in the shell, the mantle-surface continues its normal secretory 
activity, and that portion which had been in contact with the 
former septum secretes a septum anew. In short, at any given 
epoch, the creature is not secreting a tube and a septum by 
separate operations, but is secreting a shelly case about its rounded 
body, of which case one part appears to us as the continuation 
of the tube, and the other part, merging with it by indistinguishable 
boundaries, appears to us as the septum*. 

The various forms assumed by the septa in spiral shellst 
present us with a number of problems of great beauty, simple in 
their essence, but whose full investigation would soon lead us 
into mathematics of a very high order. 

We do not know in great detail how these septa are laid down; 
but the essential facts are cleart. The septum begins as a very 
thin cuticular membrane (composed apparently of a substance 
called conchyolin), which is secreted by the skin, or mantle- 
surface, of the animal; and upon this membrane nacreous matter 
is gradually laid down on the mantle-side (that is to say between 
the animal’s body and the cuticular membrane which has been 
thrown off from it), so that the membrane remains as a thin pellicle 
over the hinder surface of the septum, and so that, to begin with, 
the membranous septum is moulded on the flexible and elastic 
surface of the animal, within which the fluids of the body must 
exercise a uniform, or nearly uniform pressure. 

Let us think, then, of the septa as they would appear in their 
uncalcified condition, formed of, or at least superposed upon, an 


* “Tt has been suggested, and I think in some quarters adopted as a dogma, 
that the formation of successive septa [in Nautilus] is correlated with the recurrence 
of reproductive periods. This is not the case, since, according to my observations, 
propagation only takes place after the last septum is formed;” Willey, Zoological 
Results, p. 746, 1902. 

+ Cf. Woodward, Henry, On the Structure of Camerated Shells, Pop. Sct. Rev. 
XI, pp. 113-120, 1872. 

t See Willey, Contributions to the Natural History of the Pearly Nautilus, 
Zoological Results, etc. p. 749, 1902. Cf. also Bather, Shell-growth in Cephalopoda, 
Ann. Mag. N. H. (6), 1, pp 298-310, 1888; ibid. pp. 421-427, and other papers by 
Blake, Riefstahl, etc. quoted therein. 


XI] OF SEPTA 579 


elastic membrane. They must then follow the general law, 
applicable to all elastic membranes under uniform pressure, that 
the tension varies inversely as the radius of curvature; and we 
come back once more to our old equation of Laplace, that 


p=T(-+35). 


7 


Moreover, since the cavity below the septum is practically 
closed, and is filled either with air or with 
water, P will be constant over the whole 
area of the septum. And further, we must 
assume, at least to begin with, that the 
membrane constituting the incipient septum 
is homogeneous or isotropic. 

Let us take first the case of a straight 
cone, of circular section, more or less like an 
Orthoceras; and let us suppose that the 
septum is attached to the shell in a plane 
perpendicular to its axis. The septum itself 
must then obviously be spherical. Moreover 
the extent of the spherical surface is constant, 
and easily determined. For obviously, in 
Fig, 302, the angle LCL’ equals the sup- 
plement of the angle (LOL’) of the cone; that is to say, the 
circle of contact subtends an angle at the 
centre of the spherical surface, which is con- 
stant, and which is equal to w-—20. The 
case is not excluded where, owing to an asym- 
metry of tensions, the septum meets the side 
walls of the cone at other than a right angle, as 
in Fig. 303; and here, while the septa still 
remain portions of spheres, the geometrical 
construction for the position of their centres is 
equally easy. 

If, on the other hand, the attachment of the 
septum to the inner walls of the cone be in a> 
plane oblique to the axis, then it is evident that 
the outline of the septum will be an ellipse, and its surface an 


37—2 


Fig. 302. 


Fig. 303. 


580 THE LOGARITHMIC SPIRAL [cH. 


ellipsoid. If the attachment of the septum be not in one 
plane, but form a sinuous line of contact with the cone, then 
the septum will be a saddle-shaped surface, of great complexity 
and beauty. In all cases, provided only that the membrane be 
isotropic, the form assumed will be precisely that of a soap-bubble 
under similar conditions of attachment: that is to say, it will be 
(with the usual limitations or conditions) a surface of minimal 
area. 

If our cone be no longer straight, but curved, then the septa 
will be symmetrically deformed in consequence. A beautiful and 
interesting case is afforded us by Nautilus itself. Here the 
outline of the septum, referred to a plane, is approximately 
bounded by two elliptic curves, similar and similarly situated, 
whose areas are to one another in a definite ratio, namely as 

Ay == my — ¢—4rcota 

7 
and a similar ratio exists in Ammonites and all other close-whorled 
spirals, in which however we cannot always make the simple 
assumption of elliptical form. In a median section of Nautilus, . 
we see each septum forming a tangent to the inner and to the 
outer wall, just as it did in a section of the straight Orthoceras ; 
but the curvatures in the neighbourhood of these two points of 
contact are not identical, for they now vary inversely as the radii, 
drawn from the pole of the spiral shell. The contour of the septum 
in this median plane is a spiral curve identical with the original 
logarithmic spiral. Of this it is the “invert,” and the fact that 
the original curve and its invert are.both identical is one of the 
most beautiful properties of the logarithmic spiral*. 

But while the outline of the septum in median section is simple 
and easy to determine, the curved surface of the septum in its 
entirety is a very complicated matter, even in Nautilus which is 
one of the simplest of actual cases. For, in the first place, since 
the form of the septum, as seen in median section, is that of a 
logarithmic spiral, and as therefore its curvature is constantly 
altering, it follows that, in successive transverse sections, the 

* Tt was this that led James Bernoulli, in imitation of Archimedes, to have 


the logarithmic spiral graven on his tomb, with the pious motto, Hadem mutata 
resurgam. On Goodsir’s grave the same symbol is reinscribed. 


x1] OF SEPTA 581 


curvature is also constantly altering. But in the case of Nautilus, 
there are other aspects of the phenomenon, which we can illustrate, 
but only in part, in the following simple manner. Let us imagine 


Fig. 304. Section of Nautilus, shewing the contour of the septa in the median 
plane: the septa being (in this plane) logarithmic spirals, of which the shell- 
spiral is the evolute. 


a pack of cards, in which we have cut out of each card a similar 
concave arc of a logarithmic spiral, such as we actually see in the 
median section of the septum of a Nautilus. Then, while we hold 
the cards together, foursquare, in the ordinary position of the 


582 THE LOGARITHMIC SPIRAL [CH. 


pack, we have a simple “ruled” surface, which in any longitudinal 
section has the form of a logarithmic spiral but in any transverse 
section is a straight horizontal line. If we shear or slide the 
cards upon one another, thrusting the middle cards of the pack 
forward in advance of the others, till the one end of the pack is 
a convex, and the other a concave, ellipse, the cut edges which 
combine to represent our septum will now form a curved surface 


Fig. 305. Cast of the interior of Nautilus: to shew the contours of 
the septa at their junction with the shell-wall. 


of much greater complexity; and this is part, but not by any 
means all, of the deformation produced as a direct consequence 
of the form in Nautilus of the section of the tube within which 
the septum has to he. And the complex curvature of the surface 
will be manifested in a sinuous outline of the edge, or line of 
attachment of the septum to the tube, and will vary according 
to the configuration of the latter. In the case of Nautilus, it is 
easy to shew empirically (though not perhaps easy to demonstrate 


x1] OF SEPTA 583 


mathematically) that the sinuous or saddle-shaped form of the 
“suture” (or line of attachment of the septum to the tube) is 
such as can be precisely accounted for in this manner. It is also 
easy to see that, when the section of the tube (or “generating 
curve”) is more complicated in form, when it is flattened, grooved, 
or otherwise ornamented, the curvature of the septum and the 
outline of its sutural attachment will become very complicated 
indeed*; but it will be comparatively simple in the case of the 
first few sutures of the young shell, laid down before any overlapping 
of whorls has taken place, and this comparative simplicity of the 
first-formed sutures is a marked feature among Ammonites. 
We have other sources of complication, besides those which 
are at once introduced by the sectional form of the tube. For 
instance, the siphuncle, or little inner tube which perforates the 
septa, exercises a certain amount of tension, sometimes evidently 
considerable, upon the latter; so that we can no longer consider 
each septum as an isotropic surface, under uniform pressure ; and 
there may.be other structural modifications, or inequalities, in 
that portion of the animal’s body with which the septum is in 
contact, and by which it is conformed. It is hardly hkely, for 
all these reasons, that we shall ever attain to a full and particular 
explanation of the septal surfaces and their sutural outlines 
throughout the whole range of Cephalopod shells; but in general 
terms, the problem is probably not beyond the reach of mathe- 
matical analysis. The problem might be approached expert- 
mentally, after the manner of Plateau’s experiments, by bending 


* The “lobes” and “saddles” which arise in this manner, and on whose arrange- 
ment the modern classification of the nautiloid and ammonitoid shells largely 
depends, were first recognised and named by Leopold von Buch, Ann. Scz. Nat. 
XXVO, xxvot, 1829. 

+ Blake has remarked upon the fact (op. cit. p. 248) that in some Cyrtocerata 
we may have a curved shell in which the ornaments approximately run at a constant 
angular distance from the pole, while the septa approximate to a radial direction ; 
and that “thus one law of growth is illustrated by the inside, and another by the 
outside.” In this there is nothing at which we need wonder. It is merely a case 
where the generating curve is set very obliquely to the axis of the shell; but where 
the septa, which have no necessary relation to the mouth of the shell, take their 
places, as usual, at a certain definite angle to the walls of the tube. This relation 
of the septa to the walls of the tube arises after the tube itself is fully formed, 
and the obliquity of growth of the open end of the tube has no relation to the 
matter. 


584 THE LOGARITHMIC SPIRAL [CH. 


a wire into the complicated form of the suture-line, and studying 
the form of the liquid film which constitutes the corresponding 
surface minimae areae. ; 


Fig. 306. Ammonites (Sonninia) Sowerbyi. (From Zittel, after 
Steinmann and Déderlein.) 


In certain Ammonites the septal outline is further complicated 
in another way. Superposed upon the usual sinuous outline, with 
its “lobes” and “saddles,” we have here a minutely ramified, or 
arborescent outline, in which all the branches terminate in wavy, 


Fig. 307. Suture-line of a Triassic Ammonite (Pinacoceras). 
(From Zittel, after Hauer.) 


” 


more or less circular arcs,—looking just like the ‘landscape 
marble’ from the Bristol Rhaetic. We have no difficulty in 
recognising in this a surface-tension phenomenon. The figures 
are precisely such as we can imitate (for instance) by pouring a 


xq] CONCLUSION 585 


few drops of milk upon a greasy plate, or of oil upon an alkaline 
solution. 

We have very far from exhausted, we have perhaps little 
more than begun, the study of the logarithmic spiral and the 
associated curves which find exemplification in the multitudinous 
diversities of molluscan shells. But, with a closing word or two, 
we must now bring this chapter to an end. 

In the spiral shell we have a problem, or a phenomenon, of 
growth, immensely simplified by the fact that each successive 
increment is irrevocably fixed in regard to magnitude and position, 
instead of remaining in a state of flux and sharing in the further 
changes which the organism undergoes. In such a structure, then, 
we have certain primary phenomena of growth manifested in their 
original simplicity, undisturbed by secondary and conflicting 
phenomena. What actually grows is merely the lip of an orifice, 
where there is produced a ring of solid material, whose form we 
have treated of under the name of the generating curve; and 
this generating curve grows in magnitude without alteration of 
its form. Besides its increase in areal magnitude, the growing 
curve has certain strictly limited degrees of freedom, which define 
its motions in space: that is to say, it has a vector motion at 
right angles to the axis of the shell; and it has a sliding motion 
along that axis. And, though we may know nothing whatsoever 
about the actual velocities of any of these motions, we do know 
that they are so correlated together that their relative velocities 
remain constant, and accordingly the form and symmetry of the 
whole system remain in general unchanged. 

But there is a vast range of possibilities in regard to every 
one of these factors: the generating curve may be of various 
forms, and even when of simple form, such as an ellipse, its axes 
may be set at various angles to the system; the plane also in 
which it hes may vary, almost indefinitely, in its angle relatively 
to that of any plane of reference in the system; and in the several 
velocities of growth, of rotation and of translation, and therefore 
in the ratios between all these, we have again a vast range of 
possibilities. We have then a certain definite type, or group of 
forms, mathematically isomorphous, but presenting infinite diver- 
sities of outward appearance: which diversities, as Swammerdam 


586 THE LOGARITHMIC SPIRAL [CH. XI 


said, ex sola nascuntur diversitate gyrationum; and which accord- 
ingly are seen to have their origin in. differences of rate, or of 
magnitude, and so to be, essentially, neither more nor less than 
differences of degree. 

In nature, we find these fone presenting themselves with 
but little relation to the character of the creature by which they 
are produced. Spiral forms of certain particular kinds are common 
to Gastropods and to Cephalopods, and to diverse families of 
each; while outside the class of molluscs altogether, among the 
Foraminifera and among the worms (as in Spirorbis, Spirographis, 
and in the Dentalium-lke shell of Ditrupa), we again meet with 
similar and corresponding forms. 

Again, we find the same forms, or forms which (save for external 
ornament) are mathematically identical, repeating themselves in 
all periods of the world’s geological history; and, irrespective of 
climate or local conditions, we see them mixed up, one with 
another, in the depths and on the shores of every sea. It is hard 
indeed (to my mind) to see where Natural Selection necessarily 
enters in, or to admit that it has had any share whatsoever in the 
production of these varied conformations. Unless indeed we use 
the term Natural Selection in a sense so wide as to deprive it of 
any purely biological] significance; and so recognise as a sort of 
natural selection whatsoever nexus of causes suffices to differ- 
entiate between the likely and the unlikely, the scarce and the 
frequent, the easy and the hard: and leads accordingly, under 
the peculiar conditions, limitations and restraints which we call 
“ordinary circumstances,” one type of crystal, one form of cloud, 
one chemical compound, to be of frequent occurrence and another 
to be rare. 


CHAPTER XII 
THE SPIRAL SHELLS OF THE FORAMINIFERA 


We have already dealt in a few simple cases with the shells of 
the Foraminifera*; and we have seen that wherever the shell is 
but a single unit or single chamber, its form may be explained 
in general by the laws of surface tension: the assumption being 
that the little mass of protoplasm which makes the simple shell 
behaves as a fluad drop, the form of which is perpetuated when 
the protoplasm acquires its solid covering. Thus the spherical 
Orbulinae and the flask-shaped Lagenae represent drops in 
equilibrium, under various conditions of freedom or constraint; 
while the irregular, amoeboid body of Astrorhiza is a manifestation 
not of equilibrium, but of a varying and fluctuating distribution 
of surface energy. When the foraminiferal shell becomes multi- 
locular, the same general principles continue to hold; the growing 
protoplasm increases drop by drop, and each successive drop has 
its particular phenomena of surface energy, manifested at its fluid 
surface, and tending to confer upon it a certain place in the system 
and a certain shape of its own. 

It is characteristic and even diagnostic of this particular 
group of Protozoa (1) that development proceeds by a well-marked 
alternation of rest and of activity—of activity during which the 
protoplasm increases, and of rest during which the shell is formed ; 
(2) that the shell is formed at the outer surface of the protoplasmic 
organism, and tends to constitute a continuous or all but continuous 
covering; and it follows (3) from these two factors taken together 
that each successive increment is added on outside of and distinct 
from its predecessors, that the successive parts or chambers of 


* Cf. pp. 255, 463, etc. 


588 THE SPIRAL SHELLS [CH. 


the shell are of different and successive ages, that one part of the 
shell is always relatively new, and the rest old in various grades 
of seniority. 

The forms which we set together in the sister-group of Radio- 
laria are very differently characterised. Here the cells or vesicles 
of which each little composite organism is made up are but little 
separated, and in no way walled off, from one another; the hard 
skeletal matter tends to be deposited in the form of isolated 
spicules or of little connected rods or plates, at the angles, the 
edges or the interfaces of the vesicles; the cells or vesicles form 
a coordinated and cotemporaneous rather than a successive series. 


Fig. 308. Hastigerina sp.; to shew the “mouth.” 


In a word, the whole quasi-fluid protoplasmic body may be 
hkened to a little mass of froth or foam: that is to say, to an 
aggregation of simultaneously formed drops or bubbles, whose 
physical properties and geometrical relations are very different 
from those of a system of drops or bubbles which are formed one 
after another, each solidifying before the next is formed. 

With the actual origin or mode of development of the fora- 
miniferal shell we are now but little concerned. The main factor 
is the adsorption, and subsequent precipitation at the surface of 
the organism, of calcium carbonate,—the shell so formed being: 
interrupted by pores or by some larger interspace or “mouth” 
(Fig. 308), which interruptions we may doubtless interpret as 
being due to unequal distributions of surface energy. In many 


xit] OF THE FORAMINIFERA ‘° 589 


cases the fluid protoplasm “picks up” sand-grains and other 
foreign particles, after a fashion which we have already described 
(p. 463); and it cements these together with more or less of 
calcareous material. The calcareous shell is a crystalline structure, 
and the micro-crystals of calcium carbonate are so set that their 
little prisms radiate outwards in each chamber through the thick- 
ness of the wall:—which symmetry is subject to corresponding 
modification when the spherical chambers are more or less sym- 
metrically deformed*. 

In various ways the rounded, drop-like shells of the Fora- 
minifera, both simple and compound, have been artificially 
imitated. Thus, if small globules of mercury be immersed in 
water in which a little chromic acid is allowed to dissolve, as the 
little beads of quicksilver become slowly covered with a crystalline 
coat of mercuric chromate they assume various forms reminiscent 
of the monothalamic Foraminifera. The mercuric chromate has 
a higher atomic volume than the mercury which it replaces, and 
therefore the fluid contents of the drop are under pressure, which 
increases with the thickness of the pellicle; hence at some weak 
spot in the latter the contents will presently burst forth, so forming 
a mouth to the little shell. Sometimes a long thread is formed, 
just as in Rhabdammina linearis; and sometimes unduloid 
swellings make their appearance on such a thread, just as in 
R. discreta. And again, by appropriate modifications of the 
experimental conditions, it is possible (as Rhumbler has shewn) 
to build up a chambered shell. 

In a few forms, such as Globigerina and its close allies, the 
shell is beset during life with excessively long and delicate 
calcareous spines or needles. It is only in oceanic forms that 
these are present, because only when poised in water can such 

* Ina few cases, according to Awerinzew and Rhumbler, where the chambers are 
added on in concentric series, as in Orbitolites, we have the crystalline structure 
arranged radially in the radial walls but tangentially in the concentric ones: 
whereby we tend to obtain, on a minute scale, a system of orthogonal trajectories, 
comparable to that which we shall presently study in connection with the structure 


of bone. Cf. 8. Awerinzew, Kalkschale der Rhizopoden, Z. f. w. Z. uxxiv, 
pp. 478-490. 1903. 

t+ Rhumbler, L., Die Doppelschalen von Orbitolites und anderer Foraminiferen, 
ete., Arch. f. Protustenkunde, 1, pp. 193-296, 1902; and other papers. Also Die 
Foraminiferen der Planktonexpedition, 1, 1911, pp. 50-56. 


590 THE SPIRAL SHELLS [cH. 


delicate structures endure; in dead shells, such as we are much 
more familiar with, every trace of them is broken and rubbed 
away. The growth of these long needles is explained (as we have 
already briefly mentioned, on p. 440) by the phenomenon which 
Lehmann calls orientirte Adsorption—the tendency for a crystalline 
structure to grow by accretion, not necessarily in the outward form 
of a “crystal,” but continuing in any direction or orientation 
which has once been impressed upon it: in this case the spicular 
growth is simply in direct continuation of the radial symmetry 
of the micro-crystalline elements of the shell-wall. Over the 
surface of the shell the radiating spicules tend to occur in a 
hexagonal pattern, symmetrically grouped around the pores which 
perforate the shell. Rhumbler has suggested that this arrange- 
ment is due to diffusion-currents, forming little eddies about the 
base of the pseudopodia issuing from the pores: the idea being 
borrowed from Bénard, to whom is due the discovery of this type 
or order of vortices*. In one of Bénard’s experiments a thin 
layer of paraffin is strewn with particles of graphite, then warmed 
to melting, whereupon each little solid granule becomes the centre 
of a vortex; by the interaction of these vortices the particles tend 
to be repelled to equal distances from one another, and in the 
end they are found to be arranged in a hexagonal pattern}. The 
analogy is plain between this experiment and those diffusion 
experiments by which Leduc produces his beautiful hexagonal 
systems of artificial cells, with which we have dealt in a previous 
chapter (p. 320). 

But let us come back to the shell itself, and consider particu- 
larly its spiral form. That the shell in the Foraminifera should 
tend towards a spiral form need not surprise us; for we have 
learned that one of the fundamental conditions of the production 
of a concrete spiral is just precisely what we have here, namely 
the gradual development of a structure by means of successive 
increments superadded to its exterior, which then form part, 
successively, of a permanent and rigid structure. This condition 


* Bénard, H , Les tourbillons cellulaires. Ann. de Chim‘e (8), xxtv. 1901. Cf 
also the pattern of cilia on an Infusorian, as figured by Biitschli in Bronn’s 
Protozoa, m1, p. 1281, 1887. 

+ A similar hexagonal pattern is obtained by the mutual repulsion of floating 
magnets in Mr R. W. Wood’s experiments, Phil. Mag. xivi, pp. 162-164, 1898. 


x11] OF THE FORAMINIFERA 591 


is obviously forthcoming in the foraminiferal, but not at all in 
the radiolarian, shell. Our second fundamental condition of the 
production of a logarithmic spiral is that each successive increment 
shall be so posited and so conformed that its addition to the 
system leaves the form of the whole system unchanged. We 
have now to enquire into this latter condition; and to determine 
whether the successive increments, or successive chambers, of the 
foraminiferal shell actually constitute gnomons to the entire 
structure. 

It is obvious enough that the spiral shells of the Foraminifera 
closely resemble true logarithmic spirals. Indeed so precisely do 
_the minute shells of many Foraminifera repeat or simulate the 
spiral shells of Nautilus and its allies that to the naturalists of the 
early nineteenth century they were known as the Céphalopodes 
macroscopiques*, until Dujardin shewed that their little bodies 
comprised no complex anatomy of organs, but consisted merely 
of that slime-like organic matter which he taught us to call, 
- “sarcode,’ and which we learned afterwards from Schwann to 
speak of as “protoplasm.” 


Fig. 309. Nwmmulina antiquior, R. and V. (After V. von Mller.) 


One striking difference, however, is apparent between the shell 
of Nautilus and the little nautiloid or rotaline shells of the Fora- 
minifera: namely that the septa in these latter, and in all other 


* Cf. D’Orbigny, Alc., Tableau méthodique de la classe des Céphalopodes, Ann. 
des Sci. Nat. (1), vu, pp. 245-315, 1826; Dujardin, Félix, Observations nouvelles 
sur les prétendus Céphalopodes microscopiques, ibid. (2), m1, pp. 108, 109, 312-315, 
1835; Recherches sur les organismes inférieurs, ibid. tv, pp. 343-377, 1835, etc. 


592 THE SPIRAL SHELLS [CH. 


chambered Foraminifera, are convex outwards (Fig. 308), whereas 
they are concave outwards in Nautilus (Fig. 304) and in the rest 
of the chambered molluscan shells. The reason is_ perfectly 
simple. In both cases the curvature of the septum was deter- 
mined before it became ngid, and at a time when it had the 
properties either of a fluid film or an elastic membrane. In both 
cases the actual curvature is determined by the tensions of the 
membrane and the pressures to which it was exposed. Now it 
is obvious that the extrinsic pressure which the tension of the 
membrane has to withstand is on opposite sides in the two cases. 
In Nautilus, the pressure to be resisted is that produced by the 
growing body of the animal, lying to the outer side of the septum, 
in the outer, wider portion of the tubular shell. In the Foraminifer 
the septum at the time of its formation was no septum at all; 
it was but a portion of the convex surface of a drop—that portion 
namely which afterwards became overlapped and enclosed by the 
succeeding drop; and the curvature of the septum is concave 
towards the pressure to be resisted, which latter is imszde the 
septum, being simply the hydrostatic pressure of the fluid contents 
of the drop. The one septum is, speaking generally, the reverse 
of the other; the organism, so to speak, is outside the one and 
inside the other; and in both cases alike, the septum tends to 
assume the form of a surface of minimal area, as permitted, or as 
defined, by all the circumstances of the case. 

The logarithmic spiral is easily recognisable in typical cases* 
(and especially where the spire makes more than one visible 
revolution about the pole), by its fundamental property of con- 
tinued similarity: that is to say, by reason of the fact that the . 
big many-chambered shell is of just the same shape as the smailer 
and younger shell—which phenomenon is apparent and even 
obvious in the nautiloid Foraminifera, as in Nautilus itself: but 
nevertheless the nature of the curve must be verified by careful 
measurement, just as Moseley determined or verified it in his 


* Tt is obvious that the actual owtline of a foraminiferal, just as of a molluscan 
shell, may depart widely from a logarithmic spiral. When we say here, for short, 
that the shell 7s a logarithmic spiral, we merely mean that it is essentially related 
to one: that it can be inscribed in such a spiral, or that corresponding points 
(such, for instance, as the centres of gravity of successive chambers, or the 
extremities of successive septa) will always be found to lie upon such a spiral. 


x11] OF THE FORAMINIFERA 593 


original study of nautilus (cf. p. 518). This has accordingly been 
done, by various writers: and in the first instance by Valerian 
von Moller, in an elaborate study of Fusulina—a palaeozoic genus 
whose little shells have built up vast tracts of carboniferous 
limestone over great part of Huropean Russia*. 

In this genus a growing surface of protoplasm may be con- 
ceived as wrapping round and round a small initial chamber, in 
such a way as to produce a fusiform or ellipsoidal shell—a trans- 
verse section of which reveals the close-wound spiral coil. The 
following are examples of measurements of the successive whorls 
in a couple of species of this genus. 


F. cylindrica, Fischer IF. Bécki, v. Miller 
Breadth (in millimetres). 
Whorl Observed Calculated Observed Calculated 


I 132 — -079 — 
Il “195 198 -120 119 
iil -300 -297 -180 oie) 
IV 449 445 -264 267 
Vv = ase 396 “401 


In both cases the successive whorls are very nearly in the 
ratio of 1:1:5; and on this ratio the calculated values are 
based. 

Here is another of von Moller’s series of measurements of 
F. cylindrica, the measurements being those of opposite whorls— 
that is to say of whorls 180° apart: 


Breadth in mm. 096-117 :144 -176 - 216) «6-264 «= -323— -305 
Log. of mm. 982 -068 -158 -246 -334 -422 -509 ~ -597 
Diff. of logs. — 086 -090 -088 -088 -088 -087 -088 


The mean logarithmic difference is here -088, = log 1-225; or 
the mean difference of alternate logs (corresponding to a vector 
angle of 27, 1.e. to consecutive measurements along the same 
radius) is -176, = log 1-5, the same value as before. And this 
ratio of 1-5 between the breadths of successive whorls corresponds 
(as we see by our table on p. 534) to a constant angle of about 


* von Moller, V., Die spiral-gewundenen Foraminifera des russischen Kohlen- 
kalks, Mém. de l Acad. Imp. Sci., St Pétersbourg (7), xxv, 1878. 


T. G@. 38 


594 THE SPIRAL SHELLS [CH. 


86°, or just such a spiral as we commonly meet with in the 
Ammonites* (cf. p. 539). 

In Fusulina, and in some few other Foraminifera (cf. Fig. 
310, a), the spire seems to wind evenly on, with little or no 
external sign of the successive periods of growth, or successive 
chambers of the shell. The septa which mark off the chambers, 
and correspond to retardations or cessations in the periodicity of 
growth, are still to be found in sections of the shell of Fusulna; 
but they are somewhat irregular and comparatively inconspicuous ; 
the measurements we have just spoken of are taken without 


Fig. 310. A, Cornuspira foliacea, Phil.; B, Operculina complanata, Defr. 


reference to the segments or chambers, but only with reference 
to the whorls, or in other words with direct reference to the 


vectorial angle. 
The linear dimensions of successive chambers have been 


* As von Miller is careful to explain, Naumann’s formula for the “cyclo- 
centric conchospiral” is appropriate to this and other spiral Foraminifera, since 
we have in all these cases a central or initial chamber, approximately spherical, 
about which the logarithmic spiral is coiled (cf. Fig. 309). In species where the 
central chamber is especially large, Naumann’s formula is all the more advan- 
tageous. But it is plain that it is only required when we are dealing with 
diameters, or with radii; so long as we are merely comparing the breadths of 
successive whorls, the two formulae come to the same thing. 


x11] OF THE FORAMINIFERA 595 


measured in a number of cases. Van Iterson* has done so in 
various Miholinidae, with such results as the following: 


Triloculina rotunda, Orb. 


No. of chamber ... Mee eelo a as RO ee Ont i 8 9 10 
Breadth of chamber inn — 34 45 61 84 114 142 182 246 319 
Breadth of chamber in p, 

calculated Sas ee —h 84: 45160" 7928105) 140 187. 243 319 


Here the mean ratio of breadth of consecutive chambers may 
be taken as 1-323 (that is to say, the eighth root of 319/34); and 
the calculated values, as given above, are based on this deter- 
mination. 

Again, Rhumbler has measured the lnear dimensions of a 
number of rotaline forms, for instance Pulvinulina menardi 
(Fig. 259): in which common species he finds the mean linear 
ratio of consecutive chambers to be about 1-187. In both cases, 
and especially in the latter, the ratio is not strictly constant from 
chamber to chamber, but is subject to a small secondary fluctua- 
tion f. 

When the linear dimensions of successive chambers are in 
continued proportion, then, in order that the whole shell may 
constitute a logarithmic spiral, it is necessary that the several 
chambers should subtend equal angles of revolut@on at the pole. 
In the case of the Miliolidae this is obviously the case (Fig. 311); 
for in this family the chambers lie in two rows (Biloculina), or 
three rows (Triloculina), or in some other small number of series: 
so that the angles subtended by them are large, simple fractions 
of the circular arc, such as 180° or 120°. In many of the nautiloid 
forms, such as Cyclammina (Fig. 312), the angles subtended, 
though of less magnitude, are still remarkably constant, as we 


* Van Iterson, G., Mathem. u. mikrosk.-anat. Studien iiber Blattstellungen, nebst 
Betrachtungen iiber den Schalenbau der Miliolinen, 331 pp., Jena, 1907. 

+ Hans Przibram asserts that the linear ratio of successive chambers tends in 
many Foraminifera to approximate to 1-26, which =./2; in other words, that 
the volumes of successive chambers tend to double. This Przibram would bring 
into relation with another law, viz. that insects and other arthropods tend to 
moult, or to metamorphose, just when they double their weights, or increase their 
linear dimensions in the ratio of 1 : i) 2. (Die Kammerprogression der Foraminiferen 
als Parallele zur Hiutungsprogression der Mantiden, Arch. f. Entw. Mech. xxx1v 
p- 680, 1813.) Neither rule seems to me to be well grounded. 


38—2 


596 THE SPIRAL SHELLS [OH. 


Fig. 311. 1, 2, Miliolina pulchella, VOrb.; 8-5, M. linnaeana, @ Orb. 
(After Brady.) 


Fig. 312. Cyclammina cancellata, Brady. 


x11] OF THE FORAMINIFERA 597 


may see by Fig. 313; where the angle subtended by each chamber 
is made equal to 20°, and this diagrammatic figure is not per- 
ceptibly different from the other. In some cases the subtended 
angle is less constant; and in these it would be necessary to equate 
the several linear dimensions with the corresponding vector angles, 
according to our equation r= e’°°*, It is probable that, by so 
taking account of variations of 8, such variations of r as (according 
to Rhumbler’s measurements) Pulvinulina and other genera 
appear to shew, would be found to diminish or even to disappear. 


Fig. 313. Cyclammina sp. (Diagrammatic.) 


The law of increase by which each chamber bears a constant 
ratio of magnitude to the next may be looked upon as a simple 
consequence of the structural uniformity or homogeneity of the 
organism; we have merely to suppose (as this uniformity would 
naturally lead us to do) that the rate of increase is at each instant 
proportional to the whole existing mass. For if Vy, V,, etc., be 
the volumes of the successive chambers, let V, bear a constant 
proportion to Vy, so that V,;=qVo, and let V, bear the same 
proportion to the whole pre-existing volume: then 


Va= 9 (Vo + Vi) = 4 (Vo + GV o) = 99 (1+ 9) and V/V, =1+4. 


% 


598 THE SPIRAL SHELLS [CH. 


This ratio of 1/(1 + q) is easily shewn to be the constant ratio 
running through the whole series, from chamber to chamber; 
and if this ratio of volumes be constant, so also are the ratios 
of corresponding surfaces, and of corresponding linear dimensions, 
provided always that the successive increments, or successive 
chambers, are similar in form. 

We have still to discuss the similarity of form and the symmetry 
of position which characterise the successive chambers, and which, 
together with the law of continued proportionality of size, are the 
distinctive characters and the indispensable conditions of a series 
of “gnomons.” 

The minute size of the foraminiferal shell or at least of each 
successive increment thereof, taken in connection with the fluid 


Fig. 314. Orbulina universa, d’ Orb. 


or semi-fluid nature of the protoplasmic substance, is enough to 
suggest that the molecular forces, and especially the force of 
surface-tension, must exercise a controlling influence over the form 
of the whole structure; and this suggestion, or belief, is already 
implied in our statement that each successive increment of growing 
protoplasm constitutes a separate drop. These “drops,” partially 
concealed by their successors, but still shewing in part their 
rounded outlines, are easily recognisable in the various fora- 
miniferal shells which are illustrated in this chapter. 

The accompanying figure represents, to begin with, the spherical 
shell characteristic of the common, floating, oceanic Orbulina. 
In the specimen illustrated, a second chamber, superadded to the 


4 


XIE] OF THE FORAMINIFERA 599 


first, has arisen as a drop of protoplasm which exuded through the 
pores of the first chamber, accumulated on its surface, and spread 
over the latter till it came to rest in a position of equilibrium. 
We may take it that this position of equilibrium is determined, 
at least in the first instance, by the “law of the constant angle,” 
which holds, or tends to hold, in all cases where the free surface 
of a given liquid is in contact with a given solid, in presence of 
another liquid or a gas. The corresponding equations are pre- 
cisely the same as those which we have used in discussing the 
form of a drop (on p. 294); though some slight modification must 
be made in our definitions, inasmuch as the consideration of 
surface-tension is no longer appropriate at the solid surfaces, and 
the concept of surface-energy must take its place. Be that as it 
may, it is enough for us to observe that, in such a case as ours, 
when a given fluid (namely protoplasm) is in surface contact with 
a solid (viz. a calcareous shell), in presence of another fluid (sea- 
water), then the angle of contact, or angle by which the common 
surface (or interface) of the two liquids abuts against the solid wall, 
tends to be constant: and that being so, the drop will have a 
certain definite form, depending (¢nter alia) on the form of the 
surface with which it is in contact. After a period of rest, during 
which the surface of our second drop becomes rigid by calcification, 
a new period of growth will recur and a new drop of protoplasm 
be accumulated. Circumstances remaining the same, this new 
drop will meet the solid surface of the shell at the same angle as 
did the former one; and, the other forces at work on the system 
remaining the same, the form of the whole drop, or chamber, will 
be the same as before. 

According to Rhumbler, this “law of the constant angle” is 
the fundamental principle in the mechanical conformation of the 
foraminiferal shell, and provides for the symmetry of form as 
well as of position in each succeeding drop of protoplasm: which 
form and position, once acquired, become rigid and fixed with the 
onset of calcification. But Rhumbler’s explanation brings with 
it its own difficulties. It is by no means easy of verification, for 
on the very complicated curved surfaces of the shell it seems to 
me extraordinarily difficult to measure, or even to recognise, the 
actual angle of contact: of which angle of contact, by the way, 


600 THE SPIRAL SHELLS ‘[ou. 


but little is known, save only in the particular case where one of 
the three bodies is air, as when a surface of water is exposed to 
air and in contact with glass. It is easy moreover to see that in 
many of our Foraminifera the angle of contact, though it may be 
constant in homologous positions from chamber to chamber, is 
by no means constant at all points along the boundary of each 
chamber. In Cristellaria, for instance (Fig. 315), 1t would seem 
to be (and Rhumbler asserts that it actually is) about 90° on the 
outer side and only about 50° on the inner side of each septal 
partition; in Pulvinulina (Fig. 259), according to Rhumbler, the 
angles adjacent to the mouth are of 90°, and the opposite angles 


Fig. 315. Cristellaria_reniformis, Orb. 


are of 60°, in each chamber. For these and other similar discre- 
pancies Rhumbler would account by simply invoking the hetero- 
geneity of the protoplasmic drop: that is to say, by assuming that 
the protoplasm has a different composition and different properties 
(including a very different distribution of surface-energy), at 
points near to and remote from the mouth of the shell. Whether 
the differences in angle of contact be as great as Rhumbler takes 
them to be, whether marked heterogeneities of the protoplasm 
occur, and whether these be enough to account for the differences 
of angle, I cannot tell. But it seems to me that we had better 
rest content with a general statement, and that Rhumbler has 
taken too precise and narrow a view. 


x11] OF THE FORAMINIFERA 60] 


In the molecular growth of a crystal, although we must of 
necessity assume that each molecule settles down in a position of 
minimum potential energy, we find it very hard indeed to explain 
precisely, even in simple cases and after all the labours of modern 
erystallographers, why this or that position is actually a place of 
minimum potential. In the case of our little Foraminifer (just 
as in the case of the crystal), let us then be content to assert that 
each drop or bead of protoplasm takes up a position of minimum 
potential energy, in relation to all the circumstances of the case ; 
and let us not attempt, in the present state of our knowledge, to 
define that position of minimum potential by reference to angle 
of contact or any other particular condition of equlibrium. In 
most cases the whole exposed surface, on some portion of which 
the drop must come to rest, is an extremely complicated one, and 
the forces involved constitute a system which, in its entirety, is 
more complicated still; but from the symmetry of the case and 
the continuity of the whole phenomenon, we are entitled to believe 
that the conditions are just the same, or very nearly the same, 
time after time, from one chamber to another: as the one chamber 
is conformed so will the next tend to be, and as the one is situated 
relatively to the system so will its successor tend to be situated in 
turn. The physical law of minimum potential (including also the 
law of minimal area) is all that we need in order to explain, in 
general terms, the continued similarity of one chamber to another ; 
and the physiological law of growth, by which a continued pro- 
portionality of size tends to run through the series of successive 
chambers, impresses upon this series of similar increments the 
form of a logarithmic spiral. 

In each particular case the nature of the logarithmic spiral, 
as defined by its constant angle, will be chiefly determined by 
the rate of growth; that is to say by the particular ratio in which 
each new chamber exceeds its predecessor in magnitude. But 
shells having the same constant angle.(a) may still differ from one 
another in many ways—in the general form and relative position 
of the chambers, in their extent of overlap, and hence in the actual 
contour and appearance of the shell; and these variations must 
correspond to particular distributions of energy within the system, 
which is governed as a whole by the law of minimum potential. 


602 THE SPIRAL SHELLS [CH. 


Our problem, then, becomes reduced to that of investigating 
the possible configurations which may be derived from the succes- 
sive symmetrical apposition of similar bodies whose magnitudes 
are in continued proportion; and it is obvious, mathematically 
speaking, that the various possible arrangements all come under 
the head of the logarithmic spiral, together with the limiting cases 
which it includes. Since the difference between one such form 
and another depends upon the numerical value of certain 
coefficients of magnitude, it is plain that any one must tend to 
pass into any other by small and continuous gradations; in 
other words, that a classification of these forms must (like any 
classification whatsoever of logarithmic spirals or of any other 
mathematical curves), be theoretic or “artificial.” But we may 
easily make such an artificial classification, and shall probably 
find it to agree, more or less, with the usual methods of classification 
recognised by biological students of the Foraminifera. 

Firstly we have the typically spiral shells, which occur in 
great variety, and which (for our present purpose) we need hardly 
describe further. We may merely notice how in certain cases, 
for instance Globigerina, the individual chambers are little removed 
from spheres; in other words, the area of contact between the 
adjacent chambers is small. In such forms as Cyclammina and 
Pulvinulina, on the other hand, each chamber is greatly over- 
lapped by its successor, and the spherical form of each is lost in 
a marked asymmetry. Furthermore, in Globigerina and some 
others we have a tendency to the development of a helicoid spiral 
in space, as In so many of our univalve molluscan shells. The 
mathematical problem of how a shell should grow, under the 
assumptions which we have made, would probably find its most 
general statement in such a case as that of Globigerina, where the 
whole organism lives and grows freely poised in a medium whose 
density is little different from its own. 

The majority of spiral forms, on the other hand, are plane 
or discoid spirals, and we may take it that in these cases some 
force has exercised a controlling influence, so as to keep all the 
chambers in a plane. This is especially the case in forms like 
Rotalhia or Discorbina (Fig. 316), where the organism lives attached 
to a rock or a frond of sea-weed; for here (just as in the case of 


x11] OF THE FORAMINIFERA 603 


the coiled tubes which little worms such as Serpula and Spirorbis 
make, under similar conditions) the spiral disc is itself asymmetrical, 
its whorls being markedly flattened on their attached surfaces. 


Fig. 316. Discorbina bertheloti, d Orb. 


We may also conceive, among other conditions, the very 
curious case in which the protoplasm may entirely overspread the 
surface of the shell without reaching a position of equilibrium; 
in which case a new shell will be formed enclosing the old one, 


604. THE SPIRAL SHELLS [CH. 


whether the old one be in the form of a single, solitary chamber, © 
or have already attained to the form of a chambered or spiral 
shell. This is precisely what often happens in the case of Orbulina, 
when within the spherical shell we find a small, but perfectly 
formed, spiral “ Globigerina*.” 

The various Miliolidae (Fig. 311), only differ from the typical 
spiral, or rotaline forms, in the large angle subtended by each 
chamber, and the consequent abruptness of their inclination to 
each other. In these cases the outward appearance of a spiral 
tends to be lost; and it behoves us to recollect, all the more, 
that our spiral curve is not necessarily identical with the outline 


Fig. 317. A, Tertularia trochus, @Orb. B, 7’. concava, Karrer. 


of the shell, but is always a line drawn through corresponding 
points in the successive chambers of the latter. 

We reach a limiting case of the logarithmic spiral when the 
chambers are arranged in a straight line; and the eye will tend 
to associate with this limiting case the much more numerous forms 
in which the spiral angle is small, and the shell only exhibits a 
gentle curve with no succession of enveloping whorls. This 
constitutes the Nodosarian type (Fig. 87, p. 262); and here again, 
we must postulate some force which has tended to keep the 
chambers in a rectilinear series: such for instance as gravity, 
acting on a system of “hanging drops.” 


* Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina, Wiegmann’s 
Archiv, XLIx, p. 428, 1883; Brady, Chall. Rep., p. 607, 1884. 


xm] - OF THE FORAMINIFERA 605 


In Textularia and its allies (Fig. 317), we have a_ precise 
parallel to the helicoid cyme of the botanists (cf. p. 502): that 
is to say we have a screw translation, perpendicular to the plane 
of the underlying logarithmic spiral. In other words, in tracing 
a genetic spiral through the whole succession of chambers, we do 
so by a continuous vector rotation, through successive angles of 
180° (or 120° in some cases), while the pole moves along an axis 
perpendicular to the original plane of the spiral. | 

Another type is furnished by the “cyclic” shells of the 
Orbitolitidae, where small and numerous chambers tend to be 
added on round and round the system, so building up a circular 
flattened disc. This again we perceive to be, mathematically, a 
hmiting case of the logarithmic spiral, where the spiral has become 
a circle and the constant angle is now an angle of 90°. 

Lastly there are a certain number of Foraminifera in which, 
without more ado, we may simply say that the arrangement of 
the chambers is irregular, neither the law of constant ratio of 
magnitude nor that of constant form being obeyed. The chambers 
are heaped pell-mell upon one another, and such forms are known 
to naturalists ‘as the Acervularidae. 

While in these last we have an extreme lack of regularity, we 
must not exaggerate the regularity or constancy which the more 
ordinary forms display. We may think it hard to believe that 
the simple causes, or simple laws, which we have described should 
operate, and operate again and again, in millions of individuals to 
produce the same delicate and complex conformations. But we 
are taking a good deal for granted if we assert that they do so, 
and in particular we are assuming, with very little proof, the 
“constancy of species” in this group of animals. Just as Verworn 
has shewn that the typical Amoeba proteus, when a trace of alkali 
is added to the water in which it lives, tends, by alteration of 
surface tensions, to protrude the more delicate pseudopodia 
characteristic of A. radiosa,—and again when the water is rendered 
a little more alkaline, to turn apparently into the so-called A. 
limax,—so it is evident that a very slight modification in ,the 
surface-energies concerned, might tend to turn one so-called 
species into another among the Foraminifera. To what extent 
this process actually occurs, we do not know. 


606 THE SPIRAL SHELLS [CH. 


But that this, or something of the kind, does actually occur 
we can scarcely doubt. For example in the genus Peneroplis, the 
first portion of the shell consists of a series of chambers arranged 
in a spiral or nautiloid series; but as age advances the spiral is 
apt to be modified in various ways*. Sometimes the successive 
chambers grow rapidly broader, the whole shell becoming fan- 
shaped. Sometimes the chambers become narrower, till they no 
longer enfold the earlier chambers but only come in contact each 
with its immediate predecessor: the result being that the shell 
straightens out, and (taking into account the earlier spiral portion) 
may be described as crozier-shaped. Between these extremes of 
shape, and in regard to other variations of thickness or thinness, 
roughness or smoothness, and so on, there are innumerable 
gradations passing one into another and intermixed without regard 
to geographical distribution :—‘wherever Peneroplides abound 
this wide variation exists, and nothing can be more easy than to 
pick out a number of striking specimens and give to each a dis- 
tinctive name, but 7m no other way can they be divided into 
‘species. +’? Some writers have wondered at the peculiar 
variability of this particular shellt; but for all we know of the 
life-history of the Foraminifera, it may well be that a great 
number of the other forms which we distinguish as separate species 
and even genera are no more than temporary manifestations of 
the same variability §. 

* Cf. Brady, H. B., Challenger Rep., Foraminifera, 1884, p. 203, pl. xm. 

+ Brady, op. cit., p. 206; Batsch, one of the earliest writers on Foraminifera. 
had already noticed that this whole series of ear-shaped and crozier-shaped shells 
was filled in by gradational forms; Conchylien des Seesandes, 1791, p. 4, pl. v1, 
fig. 15 a-f. See also, in particular, Dreyer, Peneroplis; eine Studie zur biologischen 
Morphologie und zur Speciesfrage, Leipzig, 1898; also Eimer und Fickert, Artbildung 
und Verwandschaft bei den Foraminiferen, Tiibinger zool. Arbeiten, m1, p. 35, 
1899. 

t Doflein, Protozoenkunde, 1911, p. 263; “Was diese Art veranlasst in dieser 
Weise gelegentlich zu variiren, ist vorlaufig noch ganz rathselhaft.” 

§ In the case of Globigerina, some fourteen species (out of a very much larger 
number of described forms) were allowed by Brady (in 1884) to be distinct; and 
this list has been, I believe, rather added to than diminished. But these so-called 
species depend for the most part on slight differences of degree, differences in the 
angle of the spiral, in the ratio of magnitude of the segments, or in their area of 
contact one with another. Moreover with the exception of one or two “dwarf” 
forms, said to be limited to Arctic and Antarctic waters, there is no principle of 
geographical distribution to be discerned amongst them. A species found fossil 


x11] OF THE FORAMINIFERA 607 


Conclusion. 


If we can comprehend and interpret on some such lines as 
these the form and mode of growth of the foraminiferal shell, we 
may also begin to understand two striking features of the group, 
namely, on the one hand the large number of diverse types or 
families which exist and the large number of species and varieties 
within each, and on the other the persistence of forms which in 
many cases seem to have undergone little change or none at all 
from the Cretaceous or even from earlier periods to the present 
. day. In few other groups, perhaps only among the Radiolaria, 
do we seem to possess so nearly complete a picture of all possible 
transitions between form and form, and of the whole branching 
system of the evolutionary tree: as though little or nothing of it 
had ever perished, and the whole web of life, past and present, 
were as complete as ever. It leads one to imagine that these 
shells have grown according to laws so simple, so much in harmony 
with their material, with their environment, and with all the 
forces internal and external to which they are exposed, that none 
is better than another and none fitter or less fit to survive. It 
invites one also to contemplate the possibility of the lines of 
possible variation being here so narrow and determinate that 
identical forms may have come independently into being again 
and again. . 

While we can trace in the most complete and beautiful manner 
the passage of one form into another among these little shells, 
and ascribe them all at last (if we please) to a series which starts 
with the simple sphere of Orbulina or with the amoeboid body of 
Astrorhiza, the question stares us in the face whether this be an 
“evolution” which we have any right to correlate with historic 
time. The mathematician can trace one conic section into 
another, and “evolve” for example, through innumerable graded 
ellipses, the circle from the straight line: which tracing of con- 
tinuous steps is a true “evolution,” though time has no part 
therein. It was after this fashion that Hegel, and for that matter 
Aristotle himself, was an evolutionist—to whom evolution was 


in New Britain turns up in the North Atlantic: a species described from the West 
Indies is rediscovered at the ice-barrier. of the Antarctic. 


608 THE SPIRAL SHELLS ; [CH. 


a mental concept, involving order and continuity in thought, but 
not an actual sequence of events in time. Such a conception of 
evolution is not easy for the modern biologist to grasp, and harder 
still to appreciate. And so it is that even those who, like Dreyer* 
and like Rhumbler, study the foraminiferal shell as a physical 
system, who recognise that its whole plan and mode of growth is 
closely akin to the phenomena exhibited by fluid drops under 
particular conditions, and who explain the conformation of the 
shell by help of the same physical principles and mathematical 
laws—yet all the while abate no jot or tittle of the ordinary 
postulates of modern biology, nor doubt the validity and universal 
applicability of the concepts of Darwinian evolution. For these 
writers the biogenetisches Grundgesetz remains impregnable. The 
Foraminifera remain for them a great family tree, whose actual, 
pedigree is traceable to the remotest ages; in which historical 
evolution has coincided with progressive change; and in which 
structyral fitness for a particular function (or functions) has 
exercised its selective action and ensured “the survival of the 
fittest.” By successive stages of historic evolution we are supposed 
to pass from the irregular Astrorhiza to a Rhabdammina with its 
more concentrated disc; to the forms of the same genus which 
consist of but a single tube with central chamber; to those where 
this chamber is more and more distinctly segmented; so to the 
typical many-chambered Nodosariae; and from these, by another 
definite advance and later evolution to the spiral Trochamminae. 
After this fashion, throughout the whole varied series of the 
Foraminifera, Dreyer and Rhumbler (following Neumayr) recog- 
nise so many successions of related forms, one passing into another, 
and standing towards it in a definite relationship of ancestry or 
descent. Each evolution of form, from simpler to more complex, 
is deemed to have been attended by an advantage to the 
organism, an enhancement of its chances of survival or perpetua- 
tion; hence the historically older forms are, on the whole, 
structurally the simpler; or conversely the simpler forms, such 
as the simple sphere, were the first to come into being in prim- 
eval seas; and finally, the gradual development and increasing 


* Dreyer, F., Principien der Geriistbildung bei Rhizopoden, etc., Jen. Zeifschr. 
XXVI, pp. 204-468, 1892. 


x11] OF THE FORAMINIFERA 609 


complication of the individual within its own lifetime is held to 
be at least a partial recapitulation of the unknown history of 
its race and dynasty*. 

We encounter many, difficulties when we try to extend such 
concepts as these to the Foraminifera. We are led for instance 
to assert, as Rhumbler does, that the increasing complexity of the 
shell, and of the manner in which one chamber is fitted on another, 
makes for advantage; and the particular advantage on which 
Rhumbler rests his argument is strength. Increase of strength, die 
Festigkeitssteigerung, is according to him the guiding principle in 
foraminiferal evolution, and marks the historic stages of their 
development in geologic time. But in days gone by I used to 
see the beach of a little Connemara bay bestrewn with millions 
upon millions of foraminiferal shells, simple Lagenae, less simple 
Nodosariae, more complex Rotalae: all drifted by wave and 
gentle current from their sea-cradle to their sandy grave: all 
lying bleached and dead: one more delicate than another, but all 
(or vast multitudes of them) perfect and unbroken. And so I 
am not inclined to believe that niceties of form affect the case 
very much: nor in general that foraminiferal life involves a 
struggle for existence wherein breakage is a constant danger to 
be averted, and increased strength an advantage to be ensuredt. 

In the course of the same argument Rhumbler remarks that 
Foraminifera are absent from the coarse sands and gravels{, as 
Williamson indeed had observed many years ago: so averting, or 


* A difficulty arises in the case of forms (like Peneroplis) where the young shell 
appears to be more complex than the old, the first formed portion being closely 
coiled while the later additions become straight and simple: “die biformen Arten 
verhalten sich, kurz gesagt. gerade umgekehrt als man nach dem biogenetischen 
Grundgesetz erwarten sollte,” Rhumbler, op. cit., p. 33 ete. 

7 “Das Festigkeitsprinzip als Movens der Weiterentwicklung ist zu interessant 
und fiir die Autstellung meines Systems zu wichtig um die Frage unerértert zu 
lassen, warum diese Bevorziigung der Festigkeit stattgefunden hat. Meiner 
Ansicht nach lautet die Antwort auf diese Frage einfach, weil die Foraminiferen 
meistens unter Verhaltnissen leben, die ihre Schalen in hohem Grade der Gefahr 
des Zerbrechens aussetzen; es muss also eine fortwahrende Auslese des Festeren 
stattfinden,” Rhumbler, op. cit., p. 22. 

t “Die Foraminiferen kiesige oder grobsandige Gebiete des Meeresbodens 
nicht lieben, u.s.w.”: where the last two words have no particular meaning, save 
only that (as M. Aurelius says) “of things that use to be, we say commonly that 
they love to be.” 


T. G. 39 


610 THE SPIRAL SHELLS [cH. 


at least escaping, the dangers of concussion. But this is after 
all a very simple matter of mechanical analysis. The coarseness 
or fineness of the sediment on the sea-bottom is a measure of the 
current: where the current is strong the larger stones are washed 
clean, where there is perfect stillness the finest mud settles down; 
and the light, fragile shells of the Foraminifera find their appro- 
priate place, like every other graded sediment, in this spontaneous 
order of lixiviation. 

The theorem of Organic Evolution is one thing; the problem 
of deciphering the lines of evolution, the order of phylogeny, the 
degrees of relationship and consanguinity, is quite another. Among 
the higher organisms we arrive at conclusions regarding these 
things by weighing much circumstantial evidence, by dealing with 
the resultant of many variations, and by considering the probability 
or improbability of many coincidences of cause and effect; but 
even then our conclusions are at best uncertain, our judgments 
are continually open to revision and subject to appeal, and all 
the proof and confirmation we can ever have is that which comes 
from the direct, but fragmentary evidence of palaeontology*. 

But in so far as forms can be shewn to depend on the play of 
physical forces, and the variations of form to be directly due to 
simple quantitative variations in these, just so far are we thrown 
back on our guard before the biological conception of consan- 
guinity, and compelled to revise the vague canons which connect 
classification with phylogeny. 

The physicist explains in terms of the properties of matter, 
and classifies according to a mathematical analysis, all the drops 
and forms of drops and associations of drops, all the kinds of 
froth and foam, which he may discover among inanimate things; 
and his task ends there. But when such forms, such conformations 
and configurations, occur among lwing things, then at once the 
biologist introduces his concepts of heredity, of historical evolution, 
of succession in time, of recapitulation of remote ancestry in 
individual growth, of common origin (unless contradicted by 
direct evidence) of similar forms remotely separated by geo- 
graphic space or geologic time, of fitness for a function, of 


* Tn regard to the Foraminifera, “‘die Palaeontologie lasst uns leider an Anfang 
der Stammesgeschichte fast ginzlich im Stiche,” Rhumbler, op. cii., p. 14. 


xi] OF THE FORAMINIFERA 611 


adaptation to an environment, of higher and lower, of “better” 
and “worse.” This is the fundamental difference between the 
“explanations” of the physicist and those of the biologist. 

In the order of physical and mathematical complexity there is 
no question of the sequence of historic time. The forces that 
bring about the sphere, the cylinder or the ellipsoid are the same 
yesterday and to-morrow. A snow-crystal is the same to-day as 
when the first snows fell. The physical forces which mould the 
forms of Orbulina, of Astrorhiza, of Lagena or of Nodosaria to-day 
were still the same, and for aught we have reason to believe the 
physical conditions under which they worked were not appreciably 
different, in that yesterday which at call the Cretaceous epoch ; 
or, for aught we know, throughout all that duration of time which 
is marked, but not measured, by the geological record. 

In a word, the minuteness of our organism brings its conforma- 
tion as a whole within the range of the molecular forces; the 
laws of its growth and form appear to le on simple lines; what 
Bergson calls* the “ideal kinship” is plain and certain, but the 
“material affiliation” is problematic and obscure; and, in the 
end and upshot, it seems to me by no means certain that the 
biologist’s usual mode of reasoning is appropriate to the case, or 
that the concept of continuous historical evolution must necessarily, 
or may safely and legitimately, be employed. 


* The evolutionist theory, as Bergson puts it, “‘consists above all in establishing 
relations of ideal kinship, and in maintaining that wherever there is this relation of, 
so to speak, logical affiliation between forms, there is also a relation of chronological 
succession between the species in which these forms are materialised” : Creative 
Evolution, 1911, p. 26. Cf. supra, p. 251. 


39—2 


CHAPTER XIII 


THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: 
WITH A NOTE ON TORSION 


We have had so much to say on the subject of shell-spirals 
that we must deal briefly with the analogous problems which are 
presented by the horns of sheep, goats, antelopes and other 
horned quadrupeds; and all the more, because these horn-spirals 
are on the whole less symmetrical, less easy of measurement than 
those of the shell, and in other ways also are less easy of investi- 
gation. Let us dispense altogether in this case with mathematics ; 
and be content with a very simple account of the configuration 
of a horn. 

There are three types of horn which deserve separate con- 
sideration: firstly, the horn of the rhinoceros; secondly the 
horns of the sheep, the goat, the ox or the antelope, that is to say, 
of the so-called hollow-horned ruminants; and thirdly, the solid 
bony horns, or “antlers,” which are characteristic of the deer. 

The horn of the rhinoceros presents no difficulty. It is 
physiologically equivalent to a mass of consolidated hairs, and, 
like ordinary hair, it consists of non-living or “formed” material, 
continually added to by the living tissues at its base. In section, 
that is to say in the form of its “generating curve,” the horn is 
approximately elliptical, with the long axis fore-and-aft, or, in 
some species, nearly circular. Its longitudinal growth proceeds 
with a maximum velocity anteriorly, and a minimum posteriorly ; 
and the ratio of these velocities being constant, the horn curves 
into the form of a logarithmic spiral in the manner that we have 
already studied. The spiral is of small angle, but in the longer- 
horned species, such as the great white rhinoceros (Ceratorhinus), 
the spiral form is distinctly to be recognised. As the horn 


cH. x11] THE HORNS OF SHEEP AND GOATS 613 


occupies a median position on the head,—a position, that is to say, 
of symmetry in respect to the field of force on either side,—there 
is no tendency towards a lateral twist, and the horn accordingly 
develops as a plane logarithmic spiral. When two horns coexist, 
the hinder one is much the smaller of the two: which is as much 
as to say that the force, or rate, of growth diminishes as we pass 
backwards, just as it does within the limits of the single horn. 
And accordingly, while both horns have essentially the same 
shape, the spiral curvature is less manifest in the second one, 
simply by reason of its comparative shortness. 

The paired horns of the ordinary hollow-horned ruminants, 
such as the sheep or the goat, grow under conditions which are 
in some respects similar, but which differ in other and important 
respects from the conditions under which the horn grows in the 
rhinoceros. As regards its structure, the entire horn now consists 
of a bony core with a covering of skin; the inner, or dermal, 
layer of the latter is richly supplied with nutrient blood-vessels, 
while the outer layer, or epidermis, develops the fibrous or 
chitinous material, chemically and morphologically akin to a 
mass of cemented or consolidated hairs, which constitutes the 
“sheath” of the horn. A zone of active growth at the base of 
the horn keeps adding to this sheath, ring by ring, and the specific 
form of this annular zone is, accordingly, that of the “ generating 
curve” of the horn. Each horn no longer lies, as it does in the 
rhinoceros, in the plane of symmetry of the animal of which it 
forms a part; and the limited field of force concerned in the 
genesis and growth of the horn is bound, accordingly, to be more 
or less laterally asymmetrical. But the two horns are in sym- 
metry one with another; they form “conjugate” spirals, one 
being the “ mirror-image” of the other. Just as in the hairy coat 
of the animal each hair, on either side of the median “parting,” 
tends to have a certain definite direction of its own axis, inclined 
away from the median axial plane of the whole system, so is it 
both with the bony core of the horn and with the consolidated 
mass of hairs or hair-like substance which constitutes its sheath; 
the primary axis of the horn is more or less inclined to, and may 
even be nearly perpendicular to, the axial plane of the animal. 

The growth of the horny sheath is not continuous, but more or 


614 THE SHAPES OF HORNS [CH. 


less definitely periodic: sometimes, as in the sheep, this periodicity 
is particularly well-marked, and causes the horny sheath to be 
composed of a series of all but separate rings, which are supposed 
to be formed year by year, and so to record the age of the animal*. 

Just as we sought for the true generating curve in the orifice, 
or “lip,” of the molluscan shell, so we might be apt to assume 
that in the spiral horn the generating curve corresponded to the 
lip or margin of one of the horny rings or annuli. This annular 
margin, or boundary of the ring, is usually a smuous curve, not 
lying in a plane, but such as would form the boundary of an 
anticlastic surface of great complexity: to the meaning and origin 
of which phenomenon we shall return presently. But, as we have 
already seen in the case of the molluscan shell, the complexities 
of the lip itself, or of the corresponding lines of growth upon the 
shell, need not concern us in our study of the development of the 
spiral: inasmuch as we may substitute for these actual boundary 
lines, their “trace,” or projection on a plane perpendicular to the 
axis—in other words the simple outline of a transverse section 
of the whorl. In the horn, this transverse section is often circular 
or nearly so, as in the oxen and many antelopes: it now and then 
becomes of somewhat complicated polygonal outline, as in a 
highland ram; but in many antelopes, and in most of the sheep, 
the outline is that of an isosceles, or sometimes nearly equilateral 
triangle, a form which is typically displayed, for instance, in 
Ovis Ammon. The horn in this latter case is a trihedral prism, 
whose three faces are, (1) an upper, or frontal face, in continuation 
of the plane of the frontal bone; (2) an outer, or orbital, starting 
from the upper margin of the orbit; and (3) an inner, or “nuchal,” 
abutting on the parietal boney. Along these three faces, and 
their corresponding angles or edges, we can trace in the fibrous 
substance of the horn a series of homologous spirals, such as we 


* In the case of the ram’s horn, the assumption that the rings are annual is 
probably justified. In cattle they are much less conspicuous, but are sometimes 
well-marked in the cow; and in Sweden they are then called “calf-rings,” from 
a belief that they record the number of offspring. That is to say, the growth of 
the horn is supposed to be retarded during gestation, and to be accelerated after 
parturition, when superfluous nourishment seeks a new outlet. (Cf. Lénnberg, 
P.Z.S., p. 689, 1900.) 

+ Cé£. Sir V. Brooke, On the Large Sheep of the Thian Shan, P.Z.S., p. 511, 1875. 


U 


XIII | OF SHEEP AND GOATS 615 


have called in a preceding chapter the “ensemble of generating 
spirals” which constitute the surface. 

In some few cases, of which the male musk ox is one of the 
most notable, the horn is not developed in a continuous spiral 
curve. It changes its shape as growth proceeds; and this, as 
we have seen, is enough to show that it does not constitute a 
logarithmic spiral. The reason is that the bony exostoses, or 
horn-cores, about which the horny sheath is shaped and moulded, 
neither grow continuously nor even remain of constant size after 
attaining their full growth. But as the horns grow heavy the 
bony core is bent downwards by their weight, and so guides the 


Fig. 318. Diagram of Ram’s horns. (After Sir Vincent Brooke, from 
P.Z.8.) a, frontal; b, orbital; c, nuchal surface. 


growth of the horn in a new direction. Moreover as age advances, 
the horn-core is further weakened and to a great extent absorbed : 
and the horny sheath or horn proper, deprived of its support, 
continues to grow, but in a flattened curve very different from 
its original spiral*. The chamois is a somewhat analogous case. 
Here the terminal, or oldest, part of the horn is curved; it tends 
to assume a spiral form, though from its comparative shortness 
it seems merely to be bent into a hook. But later on, the bony 
core within, as it grows and strengthens, stiffens the horn, and 
guides it into a straighter course or form. The same phenomenon 


* Cf. Lénnberg, E., On the Structure of the Musk Ox, P.Z.S., pp. 686-718, 
1900. 


616 THE SHAPES OF HORNS [cH. 


of change of curvature, manifesting itself at the time when, or 
the place where, the horn is freed from the support of the internal 
core, 1s seen in a good many other antelopes (such as the hartebeest) 
and in many buffaloes; and the cases where it is most manifest 
appear to be those where the bony core is relatively short, or 
relatively weak. 


Fig. 319. Head of Arabian Wild Goat, Capra sinaitica. (After 
Sclater, from P.Z.S.) 


But in the great majority of horns, we have-no difficulty in 
recognising a continuous logarithmic spiral, nor in referring it, as 
before, to an unequal rate of growth (parallel to the axis) on two 
opposite sides of the horn, the inequality maintaining a constant 
ratio as long as growth proceeds. In certain antelopes, such as 
the gemsbok, the spiral angle is very small, or in other words 
the horn is very nearly straight; in other species of the same 
genus Oryx, such as the Beisa antelope and the Leucoryx, a gentle 


XII] OF SHEEP AND GOATS 617 


curve (not unlike though generally less than that of a Dentalium 
shell) is evident; and the spiral angle, according to the few 
measurements I have made, is found to measure from about 
20° to nearly 40°. In some of the large wild goats, such as the 
Scinde wild goat, we have a beautiful logarithmic spiral, with a 
constant angle of rather less than 70°; and we may easily arrange 
a series of forms, such for example as the Siberian ibex, the 
mouftion, Ovis Ammon, etc., and ending with the long-horned 
Highland ram: in which, as we pass from one to another, we 
recognise precisely homologous spirals, with an increasing angular 
constant, the spiral angle beg, for instance, about 75° or rather 
less in Ovis Ammon, and in the Highland ram a very little more. 
We have already seen that in the neighbourhood of 70° or 80° 
a small change of angle makes a marked difference in the appear- 
ance of the spire; and we know also that the actual length of the 
horn makes a very striking difference, for the spiral becomes 
especially conspicuous to the eye when a horn or shell is long 
enough to shew several whorls, or at least a considerable part of 
one entire whorl. 

Even in the simplest cases, such as the wild goats, the spiral 
is never (strictly speaking) a plane or discoid spiral: but in 
greater or less degree there is always superposed upon the plane 
logarithmic spiral a helical spiral in space. Sometimes the latter 
is scarcely apparent, for the helical curvature is comparatively 
small, and the horn (though long, as in the said wild goats) is not 
nearly long enough to shew a complete convolution: at other 
times, as in the ram, and still better in many antelopes, such as 
the koodoo, the helicoid or corkscrew curve of the horn is its 
most characteristic feature. 

Accordingly we may study, as in the molluscan shell, the 
helicoid component of the spire—in other words the variation in 
what we have called (on p. 555) the angle 6. This factor it is 
which, more than the constant angle of the logarithmic spiral, 
imparts a characteristic appearance to the various species of 
sheep, for instance to the various closely allied species of Asiatic 
wild sheep, or Argali. In all of these the constant angle of the 
logarithmic spiral is very much the same, but the shearing com- 
ponent differs greatly. And thus the long drawn out horns of 


618 THE SHAPES OF HORNS [OH. 


Ovis Poli, four feet or more from tip to tip, differ conspicuously 
from those of Ovis Ammon or O. hodgsoni, in which a very similar 
logarithmic spiral is wound (as it were) round a much blunter cone. 


The ram’s horn then, like the snail’s shell, is a curve of double 
curvature, in which one component has imposed upon the structure 
a plane logarithmic spiral, and the other has produced a continuous 
displacement, or “shear,” proportionate in magnitude to, and 
perpendicular or otherwise inclined in direction to, the axis of 
the former spiral curvature. The result is precisely analogous to 
that which we have studied in the snail and other spiral univalves ; 
but while the form, and therefore the resultant forces, are similar, 
the original distribution of force is not the same: for we have not 
here, as we had in the snail-shell, a “columellar” muscle, to 
introduce the component acting in the direction of the axis. We 
have, it is true, the central bony core, which in part performs an 
analogous function; but the main phenomenon here is apparently 
a complex distribution of rates of growth, perpendicular to the 
plane of the generating curve. 

Let us continue to dispense with mathematics, for the mathe- 
matical treatment of a curve of double curvature is never very 
simple, and let us deal with the matter by experiment. We have 
seen that the generating curve, or transverse section, of a typical 
ram’s horn is triangular in form. Measuring (along the curve of 
the horn) the length of the three edges of the trihedral structure 
in a specimen of Ovis Ammon, and calling them respectively the 
outer, inner, and hinder edges (from their position at the base of 
the horn, relatively to the skull), I find the outer edge to measure 
80cm., the inner 74cm., and the posterior 45cm.; let us say 
that, roughly, they are in the ratio of 9:8:5. Then, if we make 
a number of little cardboard triangles, equip each with three little 
legs (I make them of cork), whose relative lengths are as 9:8: 5, 
and pile them up and stick them all together, we straightway 
build up a curve of double curvature precisely analogous to the 
ram’s horn: except only that, in this first approximation, we have 
not allowed for the gradual increment (or decrement) of the 
triangular surfaces, that is to say, for the tapering of the horn 
due to the growth in its own plane of the generating curve. 


xu] OF SHEEP AND GOATS 619 


In this case then, and in most other trihedral or three-sided 
horns, one of the three components, or three unequal velocities of 
growth, is of relatively small magnitude, but the other two are 
nearly equal one to the other. It would involve but little change 
for these latter to become precisely equal; and again but little to 
turn the balance of inequality the other way. But the immediate 
consequence of this altered ratio of growth would be that the 
horn would appear to wind the other way, as it does in the 
antelopes, and also in certain goats, e.g. the markhor, Capra 
falconeri. 

For these two opposite directions of twist Dr Wherry has introduced a 
convenient nomenclature. When the horn winds so that we follow it frem 
base to apex in the direction of the hands of a watch, it is customary to call 
it a “left-handed” spiral. Such a spiral we have in the horn on the left-hand 
side ofaram’shead. Accordingly, Dr Wherry calls the condition homonymous, 
where, as in the sheep, a right-handed spiral is on the right side of the head, 
and a left-handed spiral on the left side; while he calls the opposite condition 
heteronymous, as we have it in the antelopes, where the right-handed twist 
is on the left side of the head, and the left-handed twist on the right-hand side. 
Among the goats, we may have either condition. Thus the domestic and 
most of the wild goats agree with the sheep; but in the markhor the twisted 
horns are heteronymous, as in the antelopes. The difference, as we have 
seen, is easily explained; and (very much as in the case of our opposite spirals 
in the apple-snail, referred to on p. 560), it has no very deep importance. 


Summarised then, in a very few words, the argument by which 
we account for the spiral conformation of the horn is as follows: 
The horn elongates by dint of continual growth within a narrow 
zone, or annulus, at its base. If the rate of growth be identical 
on all sides of this zone, the horn will grow straight; if it be 
greater on one side than on the other, the horn will become curved : 
and it probably will be greater on one side than on the other, 
because each single horn occupies an unsymmetrical field with 
reference to the plane of symmetry of the animal. If the maximal 
and minimal velocities of growth be precisely at opposite sides 
of the zone of growth, the resultant spiral will be a plane spiral: 
but if they be not precisely or diametrically opposite, then the 
spiral will be a spiral in space, with a winding or helical com- 
ponent; and it is by no means likely that the maximum and 
minimum will occur at precisely opposite ends of a diameter, for 


620 THE SHAPES OF HORNS [CH. 


no such plane of symmetry is manifested in the field of force to 
which the growing annulus corresponds or appertains. 

Now we must carefully remember that the rates of growth of 
which we are here speaking are the net rates of longitudinal 
increment, in which increment the activity of the living cells in 
the zone of growth at the base of the horn is only one (though it 
is the fundamental) factor. In other words, if the horny sheath 
were continually being added to with equal rapidity all round its 
zone of active growth, but at the same time had its elongation 
more retarded on one side than the other (prior to its complete 
solidification) by varying degrees of adhesion or membranous 
attachment to the bone core within, then the net result would be 
a spiral curve precisely such as would have arisen from initial 
inequalities in the rate of growth itself. It seems highly probable 
that this is a very important factor, and sometimes even the 
chief factor in the case. The same phenomenon of attachment to 
the bony core, and the consequent friction or retardation with 
which the sheath slides over its surface, will lead to various 
subsidiary phenomena: among others to the presence of transverse 
folds or corrugations upon the horn, and to their unequal distribu- 
tion upon its several faces or edges. And while it is perfectly true 
that nearly all the characters of the horn can, be accounted 
for by unequal velocities of longitudinal growth upon its different 
sides, it is also plain that the actual field of force is a very compli- 
cated one indeed. For example, we can easily see that (at least 
in the great majority of cases) the direction of growth of the 
horny fibres of the sheath is by no means parallel to the axis of 
the core within; accordingly these fibres will tend to wind in a 
system of helicoid curves around the core, and not only this 
helicuid twist but any other tendency to spira] curvature on the 
part of the sheath will tend to be opposed or modified by the 
resistance of the core within. But on the other hand living bone 
is a very plastic structure, and yields easily though slowly to any 
forces tending to its deformation; and so, to a considerable 
extent, the bony core itself will tend to be modelled by the curva- 
ture which the growing sheath assumes, and the final result will 
be determined by an equilibrium between these two systems of 
forces. 


xu] OF SHEEP AND GOATS 621 


While it is not very safe, perhaps, to lay down any general 
tule as to what horns are more, and what are less spirally curved, 
I think it may be said that, on the whole, the thicker the horn, 
the greater is its spiral curvature. It is the slender horns, of such 
forms as the Beisa antelope, which are gently curved, and it is 
the robust horns of goats or of sheep in which the curvature is 
more pronounced. Other things being the same, this is what we 
should expect to find; for it is where the transverse section of 
the horn is large that we may expect to find the more marked 
differences in the intensity of the field of force, whether of active 
growth or of retardation, on opposite sides or in different sectors 
thereof. 


Fig. 320. Head of Ovis Ammon, shewing St Venant’s curves. 


But there is yet another and a very remarkable phenomenon 
which we may discern in the growth of a horn, when it takes the 
form of a curve of double curvature, namely, an effect of torsional 
strain; and this it is which gives rise to the sinuous “lines of 
growth,” or sinuous boundaries of the separate horny rings, of 
which we have already spoken. It is not at first sight obvious 
that a mechanical strain of torsion is necessarily involved in the 
growth of the horn. In our experimental illustration (p. 618), we 
built up a twisted coil of separate elements, and no torsional 
strain attended the development of the system. So would it 
be if the horny sheath grew by successive annular increments, 
free save for their relation to one another, and having no attach- 
ment to the solid core within. But as a matter of fact there is 


622 THE SHAPES OF HORNS [cH. 


such an attachment, by subcutaneous connective tissue, to the 
bony core; and accordingly a torsional strain will be set up in 
the growing horny sheath, again provided that the forces of growth 
therein be directed more or less obliquely to the axis of the core; 
for a “couple” is thus introduced, giving rise to a strain which 
the sheath would not experience were it free (so to speak) to slip 
along, impelled only by the pressure of its own growth from below. 
And furthermore, the successive small increments of the growing 
horn (that is to say, of the horny sheath) are not instantaneously 
converted from living to solid and rigid substance; but there is 
an intermediate stage, probably long-continued, during which 
the new-formed horny substance in the neighbourhood of the zone 
of active growth is still plastic and capable of deformation. 
Now we know, from the celebrated experiments of St Venant*, 
that in the torsion of an elastic body, other than a cylinder of 
circular section, a very remarkable state of strain is introduced. 
If the body be thus cylindrical (whether solid or hollow), then a 
twist leaves each circular section unchanged, in dimensions and 
in figure. But in all other cases, such as an elliptic rod or a 
prism of any particular sectional form, forces are introduced which 
act parallel to the axis of the structure, and which warp each 
section into a complex anticlastic surface. Thus in the case of a 
triangular and equilateral prism, such as is shewn in section in 
Fig. 321, if the part of the rod represented in the section be twisted 
by a force acting in the direction of the arrow, then the originally 
plane section will be warped as indicated in the diagram :—where 
the full contour-lines represent elevation above, and the dotted 
lines represent depression below, the original level. On the 
external surface of the prism, then, contour-lines which were 
originally parallel and horizontal, will be found warped into smuous 
curves, such that, on each of the three faces, the curve will be 
convex upwards on one half, and concave upwards on the other 
half of the face. The ram’s horn, and still better that of Ovs 
Ammon, is comparable to such a prism, save that in section it 
is not quite equilateral, and that its three faces are not plane. 
The warping is therefore not precisely identical on the three faces 


* St Venant, De la torsion des prismes, avec des considérations sur leur flexion, 
etc., Mém. des Savants Btrangers, Paris, XIv, pp. 233-560, 1856. 


XuT] OF SHEEP AND GOATS 623 


of the horn; but, in the general distribution of the curves, it is 
in complete accordance with theory. Similar anticlastic curves 
are well seen in many antelopes; but they are conspicuous by 
their absence in the cylindrical horns of oxen. 

The better to illustrate this phenomenon, the nature of which 
is indeed obvious enough from a superficial examination of the 
horn, I made a plaster cast of one of the horny rings in a horn of 
Ovis Ammon, so as to get an accurate pattern of its sinuous edge: 
and then, filling the mould up with wet clay, I modelled an anti- 
clastic surface, such as to correspond as nearly as possible with 
the sinuous outline*. Finally, after making a plaster cast of this 
sectional surface, I drew its contour-lines (as shewn in Fig. 322), 
with the help of a simple form of spherometer. It will be seen 
that in great part this diagram is precisely similar to St Venant’s 


Fig. 321. 


diagram of the cross-section of a twisted triangular prism; and 
this is especially the case in the neighbourhood of the sharp angle 
of our prismatic section. That in parts the diagram is somewhat 
asymmetrical is not to be wondered at: and (apart from inac- 
curacies due to the somewhat rough means by which it was made) 
this asymmetry can be sufficiently accounted for by anisotropy 
of the material, by inequalities in thickness of different parts of 
the horny sheath, and especially (I think) by unequal distributions 
of rigidity due to the presence of the smaller corrugations of the 


—* This is not difficult to do, with considerable accuracy, if the clay be kept 
well wetted, or semi-fluid, and the smoothing be done with a large wet brush. 


624 THE SHAPES OF HORNS fox. 


horn. It is apparently on account of these minor corrugations 
that, in such horns as the Highland ram’s, where they are strongly 
marked, the main St Venant effect is not nearly so well shewn as 
in the smoother horns such as those of O. Ammon and its immediate 
congeners*. 


A further Note upon Torsion. 


The phenomenon of torsion, to which we have been thus 
introduced, opens up many wide questions in connection with 
form. Some of the associated phenomena are admirably illustrated 
in the case of climbing plants; but we can only deal with these 
still more briefly and parenthetically. 

The subject of climbing plants has been elaborately dealt 
with not only in Darwin’s books7, but also by a very large number 
of earlier and later writers. In “twining” plants, which constitute 
the greater number of “climbers,” the essential phenomenon is a 
tendency of the growing shoot to revolve about a vertical axis— 
a tendency long ago discussed and investigated by such writers 
as Palm, H. von Mohl and Dutrochet{. This tendency to revolu- 
tion—“ circumvolution,” as Darwin calls it, “revolving nutation,” 
as Sachs puts it—is very closely comparable to the process by which 
an antelope’s horn (such as the koodoo’s) grows into its spiral 
or rather helicoid form; and it is simply due, m like manner, to 
inequalities in the rate of growth on different sides of the growing 
stem. There is only this difference between the two cases, that 
in the antelope’s horn the zone of active growth is confined to 
the base of the horn, while in the climbing stem the same 
phenomenon is at work throughout the whole length of the growmg 
structure. This growth is in the main due to “turgescence,” 
that is to the extension, or elongation, of ready-formed cells 
through the imbibition of water; it is a phenomenon due to 
osmotic pressure. The particular stimuli to which these move- 
ments (that is to say, these inequalities of growth) have been 


* The curves are well shewn in most of Sir V. Brooke’s figures of the various 
species of Argali, in the paper quoted on p. 614. 
~ 6 Ff Climbing Plants, 1865 (2nd edit. 1875); Power of Movement in Plants, 1880. 
t Palm, Ueber das Winden der Pflanzen, 1827; von Mohl, Bau und Winden 
der Ranken, ete., 1827; Dutrochet, Mouvements révolutifs spontanés, C.R. 1843, 
etc. 


xu] A NOTE UPON TORSION 625 


ascribed, such as contact (thigmotaxis), exposure to light 
(heliotropism), and so forth, need not be discussed here*. 

A simple stem growing upright in the dark, or in uniformly 
diffused light, would be in a position of equilibrium to a field of 
force radially symmetrical about its vertical axis. But this 
complete radial symmetry will not often occur; and the radial 
anomalies may be such as arise intrinsically from structural 
peculiarities in the stem itself, or externally to it by reason of 
unequal illumination or through various other localised forces. 
The essential fact, so far as we are concerned, is that in twining 
plants we have a very marked tendency to inequalities in longi- 
tudinal growth on different aspects of the stem—a tendency which 
is but an exaggerated manifestation of one which is more or less 
present, under certain conditions, in all plants whatsoever. Just 
as in the case of the ruminants’ horns so we find here, that this 
inequality may be, so to speak, positive or negative, the maximum 
lying to the one side or the other of the twining stem; and so it 
comes to pass that some climbers twine to the one side and some 
to the other: the hop and the honeysuckle following the sun, 
and the field-convolvulus twining in the reverse direction; there 
are also some, like the woody nightshade (Solanum Dulcamara) 
which twine indifferently either way. 

Together with this circumnutatory movement, there is very 
generally to be seen an actual forsion of the twining stem—a 
twist, that is to say, about its own axis; and Mohl made the 
curious observation, confirmed by Darwin, that when a stem 
twines around a smooth cylindrical stick the torsion does not take 
place, save “only in that degree which follows as a mechanical 
necessity from the spiral winding”: but that stems which had 
climbed around a rough stick were all more or less, and generally 
much, twisted. Here Darwin did not refrain from introducing 
that teleological argument which pervades his whole train of 
reasoning: “The stem,” he says, “probably gains rigidity by 
being twisted (on the same principle that a much twisted rope 


* Cf. (e.g.) Lepeschkin, Zur Kenntnis des Mechanismus, der Variationsbewe- , 
gungen, Ber. d. d. Bot. Gesellsch. xxvi A, pp. 724-735, 1908; also A. Trondle, Der 
Einfluss des Lichtes auf die Permeabilitat des Plasmahaut, Jahrb. wiss. Bot. 
XLvu1, pp. 171-282, 1910. 


T. G. 40 


626 THE SHAPES OF HORNS [CH. 


is stiffer than a slackly twisted one), and is thus indirectly 
benefited so as to be able to pass over inequalities in its spiral 
ascent, and to carry its own weight when allowed to revolve 
freely.” The mechanical explanation would appear to be very 
simple, and such as to render the teleological hypothesis un- 
necessary. In the case of the roughened support, there is a 
temporary adhesion or “clinging” between it and the growing 
stem which twines around it; and a system of forces is thus set 
up, producing a “couple,” just as it was in the case of the ram’s 
or antelope’s horn through direct adhesion of the bony core to 
the surrounding sheath. The twist is the direct result of this 
couple, and it disappears when the support is so smooth that no 
such force comes to be exerted. 

Another important class of climbers includes the so-called 
“leaf-climbers.” In these, some portion of the leaf, generally the 
petiole, sometimes (as in the fumitory) the elongated midrib, 
curls round a support; and a phenomenon of like nature occurs 
in many, though not all, of the so-called “tendril-bearers.”’ 
Except that a different part of the plant, leaf or tendril instead of 
stem, is concerned in the twining process, the phenomenon here 
is strictly analogous to our former case; but in the resulting 
helix there is, as a rule, this obvious difference, that, while the 
twining stem, for instance of the hop, makes a slow revolution 
about its support, the typical leaf-climber makes a close, firm 
coil: the axis of the latter is nearly perpendicular and parallel 
to the axis of its support, while in the twining stem the angle 
between the two axes is comparatively small. Mathematically 
speaking, the difference merely amounts to this, that the com- 
ponent in the direction of the vertical axis is large in the one 
case, and the corresponding component is small, if not absent, 
in the other; in other words, we have in the climbing stem a 
considerable vertical component, due to its own tendency to grow 
in height, while this longitudinal or vertical extension of the 
whole system is not apparent, or little apparent, in the other 
cases. But from the fact that the twining stem tends to run 
- obliquely to its support, and the coiling petiole of the leaf-climber 
tends to run transversely to the axis of its support, there 
immediately follows this marked difference, that the phenomenon 


XIII] A NOTE UPON TORSION 627 


of torsion, so manifest in the former case, will be absent in the 
latter. 


There is one other phenomenon which meets us in the twining 
and twisted stem, and which is doubtless illustrated also, though 
not so well, in the antelope’s horn; it is a phenomenon which 
forms the subject of a second chapter of St Venant’s researches on. 
the effects of torsional strain in elastic bodies. We have already 
seen how. one effect of torsion, in for instance a prism, is to 
produce strains parallel to the axis, elevating parts and depressing 
other parts of each transverse section. But in addition to this, 
the same torsion has the effect of materially altering the form of 
the section itself, as we may easily see by twisting a square or 
oblong piece of india-rubber. If we start with a cylinder, such as 
a round piece of catapult india-rubber, and twist it on its own 
long axis, we have already seen that it suffers no other distortion ; 
it still remains a cylinder, that is to say, it is still in section every- 
where circular. But if it be of any other shape than cylindrical 
the case is quite different, for now the sectional shape tends to 
alter under the strain of torsion. Thus, if our rod be elliptical 
in section to begin with, it will, under torsion, become a more 
elongated ellipse; if it be square, its angles will become more 
prominent, and its sides will curve. inwards, till at length the 
square assumes the appearance of a four-poited star, with 
rounded angles. Furthermore, looking at the results of this 
process of modification, we find experimentally that the resultant 
figures are more easily twisted, less resistant to torsion, than 
were those from which we evolved them; and this is a very 
curious physical or mathematical fact. So a cylinder, which is 
especially resistant to torsion, is very easily bent or flexed; while 
projecting ribs or angles, such as an engineer makes in a bar or 
pillar of iron for the purpose of greatly increasing its strength in 
the way of resistance to bending, actually make it much weaker 
than before (for the same amount of metal per unit length) in the 
way of resistance to torsion. 

In the hop itself, and in a very Eoneiieeea number of other 
twining and twisting stems, the ribbed or channelled form of the 
stem is a conspicuous feature. We may safely take it, (1) that 


40—2 


628 THE SHAPES OF HORNS [CH. 


such stems are especially susceptible of torsion; and (2) that the 
effect of torsion will be to intensify any such peculiarities of 
sectional outline which they may possess, though not to initiate 
them in an originally cylindrical structure. In the leaf-climbers 
the case does not present itself, for there, as we have seen, torsion 
itself is not, or is very slightly, manifested. There are very 
distinct traces of the phenomenon in the horns of certain antelopes, 
but the reason why it is not a more conspicuous feature of the 
antelope’s horn or of the ram’s is apparently a very simple one: 
namely, that the presence of the bony core within tends to check 
that deformation which is perpendicular, while it permits that 
which is parallel, to the axis of the horn. 


Of Deer's Antlers. 


But’ let us return to our subject of the shapes of horns, and 
consider briefly our last class of these structures, namely the bony 
antlers of the various species of elk and deer*. The problems 
which these present to us are very different from those which we 
have had to do with in the antelope or the sheep. 

With regard to its structure, it is plain that the bony antler 
corresponds, upon the whole, to the bony core of the antelope’s 
horn; while in place of the hard horny sheath of the latter, we 
have the soft “velvet,” which every season covers the new growing 
antler, and protects the large nutrient blood-vessels by help of 
which the antler grows+. The main difference lies in the fact 
that, in the one case, the bony core, imprisoned within its sheath, 
is rendered incapable of branching and incapable also of lateral 
expansion, and the whole horn is only permitted to grow in length, 
while retaining a sectional contour that is identical with (or but 
little altered from) that which it possesses at its growing base: 

* For an elaborate study of antlers, see Rérig, A., Arch. f. Entw. Mech. x, 
pp. 525-644, 1900, x1, pp. 65-148, 225-309, 1901; Hoffmann, C., Zur Morphologie 
der rezenten Hirschen, 75 pp., 23 pls., 1901: also Sir Victor Brooke, On the 
Classification of the Cervidae, P.Z.S., pp. 883-928, 1878. For a discussion of the 
development of horns and antlers, see Gadow, H., P.Z.S., pp. 206-222, 1902, and 
works quoted therein. 

7+ Cf. Rhumbler, L., Ueber die Abhaingigkeit des Geweihwachstums der Hirsche, 


speziell des Edelhirsches, vom Verlauf der Blutgefaisse im Kolbengeweih, Zeitschr. 
f. Forst. und Jagdwesen, 1911, pp. 295-314. 


x11] OF THE ANTLERS OF DEER. 629 


but in the antler, on the other hand, no such restraint is imposed, 
and the living, growing fabric of bone may expand into a broad 
flattened plate over which the blood-vesselsrun. In the immediate 
neighbourhood of the main blood-vessels growth will be most 
active; in the interspaces between, it may wholly fail: with the 
result that we may have great notches cut out of the flattened 
plate, or may at length find it reduced to the form of a simple 
branching structure. The main point, as it seems to me, is that 
the “horn” is essentially an axial rod, while the “antler” is 
essentially an outspread surface*. In other words, I believe that 


Fig. 323. Antlers of Swedish Elk. (After Lénnberg, from P.Z.S.) 


the whole configuration of an antler is more easily understood by 
conceiving it as a plate or a surface, more and more notched and 
scolloped till but a slender skeleton may remain, than to look 
upon it the other way, namely as an axial stem (or beam) giving 


* The fact that in one very small deer, the little South American Coassus, the 
antler is reduced to a simple short spike, does not preclude the general distinction 
which I have drawn. In Coassus we have the beginnings of an antler, which has 
not yet manifested its tendency to expand; and in the many allied species of the 
American genus Cariacus, we find the expansion manifested in various simple 
modes of ramification or bifurcation. (Cf. Sir V. Brooke, Classification of the 
Cervidae, p. 897.) 


GaON ie THE SHAPES OF HORNS [cH. 


off branches (or tines), the interspaces between which latter may 
sometimes be filled up to form a continuous plate. 

In a sense it matters very little whether we regard the broad 
plate-like antlers of the elk or the slender branching antlers of the 
stag as the more primitive type; for we are not concerned here 
with the question of hypothetical phylogeny. And even from the 
mathematical point of view it makes little or no difference whether 
we describe the plate as constituted by the interconnection of 


Fig. 324. Head and antlers of a Stag (Cervus Duvauceli). (After 
Lydekker, from P.Z.S.) 


the branches, or the branches derived by a process of notching 
or incision from the plate. The important point for us is to 
recognise that (save for occasional slight irregularities) the 
branching system in the one conforms essentially to the curved 
plate or surface which we see plainly in the other. In short the 
arrangement of the branches is more or less comparable to that 
of the veins in a leaf, or to that of the blood-vessels as they course 
over the curved surface of an organ. It is a process of ramifica- 
tion, not, like that of a tree, in various planes, but strictly limited 


XIIT] OF THE ANTLERS OF DEER 631 


to a single surface. And just as the veins within a leaf are not 
necessarily confined (as they happen to be in most ordinary 
leaves) to a plane surface, but, as in the petal of a tulip or the 
capsule of a. poppy, may have to run their course within a curved 
surface, so does the analogy of the leaf lead us directly to the 
mode of branching which is characteristic of the antler. The 
surface to which the branches of the antler tend to be confined 
is a more or less spheroidal, or occasionally an ellipsoidal one; 
and furthermore, when we inspect any well-developed pair of 
antlers, such as those of a red deer, a sambur or a wapiti, we have 
no difficulty in seeing that the two antlers make up between them 
a single surface, and constitute a symmetrical figure, each half 
being the mirror-image of the other. 

To put the case in another way, a pair of antlers (apart from 
occasional slight irregularities) tends to constitute a figure such 
that we could conceive an elastic sheet stretched over or round 
the entire system, so as to form one continuous and even surface ; 
and not only would the surface curvature be on the whole smooth 
and even, but the boundary of the surface would also tend to be 
an even curve: that is to say the tips of all the tines would 
approximately have their locus in a continuous. curve. 

It follows from this that if we want to make a simple model of 
a set of antlers, we shall be very greatly helped by taking some 
appropriate spheroidal surface as our groundwork or scaffolding. 
The best form of surface is a matter for trial and investigation in 
each particular case; but even in a sphere, by selecting appropriate 
areas thereof, we can obtain sufficient varieties of surface to meet 
all ordinary cases. With merely a bit of sculptor’s clay or plas- 
ticine, we should be put hard to it to model the horns of a wapiti 
or a reindeer: but if we start with an orange (or a round florence 
flask) and lay our little tapered rolls of plasticine upon it, in simple 
natural curves, it is surprismg to see how quickly and successfully 
we can imitate one type of antler after another. In doing so, 
we shall be struck by the fact that our model may vary in its 
mode of branching within very considerable limits, and yet look 
perfectly natural. For the same wide range of variation is charac- 
teristic of the natural antlers themselves. As Sir V. Brooke says 
(op. cit. p. 892), “No two antlers are ever exactly alike; and the 


632 THE SHAPES OF TEETH [CH. 


variation to which the antlers are subject is so great that in the 
absence of a large series they would be held to be indicative of 
several distinct species*.” But all these many variations le 
within a limited range, for they are all subject to our general 
rule that the entire structure is essentially confined to a single 
curved surface. 

It is plain that in the curvatures both of the beam and of its 
tines, in the angles by which these latter meet the beam, and in 
the contours of the entire system, there are involved many elegant 
mathematical problems with which we cannot at present attempt 
to deal. Nor must we attempt meanwhile to enquire into the 
physical meaning or origin of these phenomena, for as yet the clue 
seems to be lacking and we should only heap one hypothesis upon 
another. That there is a complete contrast of mathematical 
properties between the horn and the antler is the main lesson with 
which, in the meantime, we must rest content. 


Of Teeth, and of Beak and Claw. 


In a fashion similar to that manifested in the shell or the 
horn, we find the logarithmic spiral to be implicit in a great many 
other organic structures where the phenomena of growth proceed 
in a similar way: that is to say, where about an axis there is some ~ 
asymmetry leading to unequal rates of longitudinal growth, and 
where the structure is of such a kind that each new increment is 
added on as a permanent and unchanging part of the entire 
conformation. Nail and claw, beak and tooth, all come under 
this category. The logarithmic spiral always tends to manifest 
itself in such structures as these, though it usually only attracts 
our attention in elongated structures, where (that is to say) the 
radius vector has described a considerable angle. When the 
canary-bird’s claws grow long from lack of use, or when the 
incisor tooth of a rabbit or a rat grows long by reason of an injury 
to the opponent tooth against which it was wont to bite, we know 
that the tooth or claw tends to grow into a spiral curve, and we 
speak of it as a malformation. But there has been no funda- 
mental change of form, save only an abnormal increase in length; 


* Cf. also the immense range of variation in elks’ horns, as described by 
Lénnberg, P.Z.S. 1, pp. 352-360, 1902. 


xm] AND OF BEAK AND CLAW 633 


the elongated tooth or claw has the selfsame curvature that it had 
when it was short, but the spiral curvature becomes more and more 
‘manifest the longer it grows. A curious analogous case is that 
of the New Zealand huia bird, in which the beak of the female 
is described as being comparatively short and straight, while that 
of the male is long and curved; it is easy to see that there is a 
slight curvature also in the beak of the female, and that the beak 
of the male shows nothing but the same curve produced. In the 
case of the more curved beaks, such as those of an eagle or a parrot, 
we may, if we please, determine the constant angle of the loga- 
rithmic spiral, just as we have done in the case of the Nautilus 
shell; and here again, as the bird grows older or the beak longer, 
the spiral nature of the curve becomes more and more apparent, 
as in the hooked beak of an old eagle, or as in the great beak of 
some large parrot such as a hyacinthine macaw. 

Let us glance at one or two instances to illustrate the spiral 
curvature of teeth. 

A dentist knows that every tooth has a curvature of its own, 
and that in pulling the tooth he must follow the direction of the 
curve; but in an ordinary tooth this curvature is scarcely visible, 
and is least so when the diameter of the tooth is large compared 
with its length. 

In the simply formed, more or less conical teeth, such as are 
those of the dolphin, and in the more or less similarly shaped canines 
and incisors of mammals in general, the curvature of the tooth 
is particularly well seen. We see it in the little teeth of a hedge- 
hog, and in the canines of a dog or a cat it is very obvious indeed. 
When the great canine of the carnivore becomes still further 
enlarged or elongated, as in Machairodus, it grows into the 
strongly curved sabre-tooth of that great extinct tiger. In rodents, 
itis the incisors which undergo a great elongation; their rate of 
growth differs, though but slightly, on the two sides, anterior and 
posterior, of the axis, and by summation of these slight differences 
in the rapid growth of the tooth an unmistakeable logarithmic 
spiral is gradually built up. We see it admirably in the beaver, 
or in the great ground-rat, Geomys. The elephant is a similar 
case, save that the tooth, or tusk, remains, owing to comparative 
lack of wear, in a more perfect condition. In the rodent (save 
only in those abnormal cases mentioned on the last page) the 


634 THE SHAPES OF TUSKS [CH. XIII 


anterior, first-formed, part of the tooth wears away as fast as it 
is added to from behind; and in the grown animal, all those 
portions of the tooth near to the pole of the logarithmic spiral 
have long disappeared. In the elephant, on the other hand, we 
see, practically speaking, the whole unworn tooth, from point to 
root; and its actual tip nearly coincides with the pole of the 
spiral. If we assume (as with no great inaccuracy we may do) 
that the tip actually coincides with the pole, then we may very 
easily construct the continuous spiral of which the existing tusk 
constitutes a part; and by so doing, we see the short, gently 
curved tusk of our ordinary elephant growing gradually into the 
spiral tusk of the mammoth. No doubt, just as in the case of 
our molluscan shells, we have a tendency to variation, both 
individual and specific, in the constant angle of the spiral; some 
elephants, and some species of elephant, undoubtedly have a 
higher spiral angle than others. But in most cases, the angle 
would seem to be such that a spiral configuration would become 
very manifest indeed if only the tusk pursued its steady growth, 
unchanged otherwise in form, till it attained the dimensions 
which we meet with in the mammoth. In a species such as 
Mastodon angustidens, or M. arvernensis, the specific angle is 
low and the tusk comparatively straight; but the American 
mastodons and the existing species of elephant have tusks which 
do not differ appreciably, except in size, from the great spiral 
tusks of the mammoth, though from their comparative shortness 
the spiral is little developed and only appears to the eye as a 
gentle curve. Wherever the tooth is very long indeed, as in the 
mammoth or the beaver, the effect of some slight and all but 
inevitable lateral asymmetry in the rate of growth begins to shew 
itself: in other words, the spiral is seen to lie not absolutely in 
a plane, but to be a curve of double curvature, like a twisted 
horn. We see this condition very well in the huge canine tusks 
of the Babirussa; it is a conspicuous feature in the mammoth, 
and it is more or less perceptible in any large tusk of the ordinary 
elephants. 

The form of a molar tooth, which is essentially a branching or 
budding system, and in which such longitudinal growth as gives 
rise to a spiral curve is but little manifest, constitutes an entirely 
different problem with which I shall not at present attempt to deal. 


CHAPTER XIV 
ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS 


The beautiful configurations produced by the orderly arrange- 
ment of leaves or florets on a stem have long been an object of 
admiration and curiosity. Leonardo da Vinci would seem, as Sir 
Theodore Cook tells us, to have been the first to record his thoughts 
upon this subject; but the old Greek and Egyptian geometers 
are not likely to have left unstudied or unobserved the spiral 
traces of the leaves upon a palm-stem, or the spiral curves of the 
petals of a lotus or the florets in a sunflower. 

The spiral leaf-order has been regarded by many learned 
botanists as involving a fundamental law of growth, of the deepest 
and most far-reaching importance; while others, such as Sachs, 
have looked upon the whole doctrine of “ phyllotaxis” as ‘‘a sort 
of geometrical or arithmetical playing with ideas,” and “the 
spiral theory as a mode of view gratuitously introduced into the 
plant.” Sachs even goes so far as to declare this doctrine “in 
direct opposition to scientific investigation, and based upon the 
idealistic direction of the Naturphilosophie,’—the mystical biology 
of Oken and his school. 

The essential facts of the case are not difficult to understand ; 
but the theories built upon them are so varied, so conflicting, and 
sometimes so obscure, that we must not attempt to submit them 
to detailed analysis and criticism. There are two chief ways by 
which we may approach the question, according to whether we 
regard, as the more fundamental and typical, one or other of the 
two chief modes in which the phenomenon presents itself. That 
is to say, we may hold that the phenomenon is displayed in its 
essential simplicity by the corkscrew spirals, or helices, which 
-mark the position of the leaves upon a cylindrical stem or on an 


636 ON LEAF-ARRANGEMENT [CH. 


elongated fir-cone; or, on the other hand, we may be more 
attracted by, and regard as of greater importance, the logarithmic 
spirals which we trace in the curving rows of florets in the discoidal 
inflorescence of a sunflower. Whether one way or the other be 
the better, or even whether one be not positively correct and the 
other radically wrong, has been vehemently debated. In my 
judgment they are, both mathematically and biologically, to be 
regarded as inseparable and correlative phenomena. 

The helical arrangement (as in the fir-cone) was carefully 
studied in the middle of the eighteenth century by the celebrated 
Bonnet, with the help of Calandrini, the mathematician. Memoirs 
published about 1835, by Schimper and Braun, greatly amplified 
Bonnet’s investigations, and introduced a nomenclature which 
still holds its own in botanical textbooks. Naumann and the 
brothers Bravais are among those who continued the investigation 
in the years immediately following, and Hofmeister, in 1868, gave 
an admirable account and summary of the work of these and 
many other writers*. 

Starting from some given level and proceeding upwards, let 
us mark the position of some one leaf (A) upon a cylindrical stem. 
Another, and a younger leaf (B) will be found standing at a certain 
distance around the stem, and a certain distance along the stem, 


* Besides papers referred to below, and many others quoted in Sach’s Botany 
and elsewhere, the following are important: Braun, Alex., Vergl. Untersuchung 
iiber die Ordnung der Schuppen an den Tannenzapfen, etc., Verh. Car. Leop. 
Akad. xv, pp. 199-401, 1831; Dr C. Schimper’s Vortrage iiber die Méglichkeit 
eines wissenschaftlichen Verstindnisses der Blattstellung, etc., Flora, xvi, pp. 145 
—191, 737-756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe 
peripherische Blattgebilde, Verhandl. Schweiz. Ges., pp. 113-117, 1836; Bravais, 
L. and A., Essai sur la disposition des feuilles curvisériées, Ann. Sci. Nat. (2), 
vil, pp. 42-119, 1837; Sur la disposition symmétrique des inflorescences, ibid., 
pp. 193-221, 291-348, vim, pp. 11-42, 1838; Sur la disposition générale des feuilles 
rectisériées, ibid. xm, pp. 5-41, 65-77, 1839; Zeising, Normalverhdltniss der 
chemischen und morphologischen Proportionen, Leipzig, 1856; Naumann, C. F., 
Ueber den Quincunx als Gesetz der Blattstellung bei Sigillaria, etc., Newes Jahrb. 
f. Miner. 1842, pp. 410-417; Lestiboudois, T., Phyllotaxie anatomique, Paris, 1848; 
Henslow, G., Phyllotavis, London, 1871; Wiesner, Bemerkungen iiber rationale 
und irrationale Divergenzen, Flora, Lvmt, pp. 113-115, 139-143, 1875; Airy, H., 
On Leaf Arrangement, Proc. R. S. xxt, p. 176, 1873; Schwendener, 8., Mechanische 
Theorie der Blattstellungen, Leipzig, 1878; Delpino, F., Causa meccanica: della 
filotassi quincunciale, Genova, 1880; de Candolle, C., Etude de Phyllotaxie, Genéve, 
1881. 


XIV] OR PHYLLOTAXIS 637 


from the first. The former distance may be expressed as a 
fractional “divergence” (such as two-fifths of the circumference 
of the stem) as the botanists describe it, or by an “angle of 
azimuth” (such as dé = 144°) as the mathematician would be more 
likely to state it. The position of B relatively to A must be 
determined, not only by this angle ¢, in the horizontal plane, but 
also by an angle (@) in the vertical plane; for the height of B above 
the level of A, in comparison with the diameter of the cylinder, 
will obviously make a great difference in the appearance of the 
whole system, in short the position of each leaf must be expressed 
by F(¢.sin@). But this matter botanical students have not 
concerned themselves with; in other words, their studies have 
been limited (or mainly limited) to the relation of the leaves to 
one another in azimuth. 

Whatever relation we have found between A and B, let 
pretisely the same relation subsist between B and C: and so on. 
Let the growth of the system, that is to say, be continuous and 
uniform ; it is then evident that we have the elementary conditions 
for the development of a simple cylindrical helix; and this 
“primary helix” or “genetic spiral” we can now trace, winding 
-round and round the stem, through A, B, C, ete. But if we can 
trace such a helix through 4, B, C, it follows from the symmetry 
of the system, that we have only to join A to some other leaf to 
trace another spiral helix, such, for instance, as A, C, EH, etc.; 
parallel to which will run another and similar one, namely in this 
case B, D, F, etc. And these spirals will run in the opposite 
direction to the spiral ABC. 

In short, the existence of one helical arrangement of points 
implies and involves the existence of another and then another 
helical pattern, just as, in the pattern of a wall-paper, our eye 
travels from one linear series to another. . 

A modification of the helical system will be introduced when, 
instead of the leaves appearing, or standing, in singular succession, 
we get two or more appearing simultaneously upon the same level. 
If there be two such, then we shall have two generating spirals 
precisely equivalent to one another; and we may call them 
A, B, C, etc., and A’, B’, C’, and so on. These are the cases 
which we call “whorled” leaves, or in the simplest case, where 


638 ON LEAF-ARRANGEMENT [CH. 


the whorl consists of two opposite leaves only, we call them 
decussate. : 


Among the phenomena of phyllotaxis, two points in particular 
have been found difficult of explanation, and have aroused dis- 
cussion. These are (1), the presence of the logarithmic spirals 
such as we have already spoken of in the sunflower; and (2) the 
fact that, as regards the number of the helical or spiral rows, 
certain numerical coincidences are apt to recur again and again, 
to the exclusion of others, and so to become characteristic features 
of the phenomenon. 

' The first of these appears to me to present no difficulty. It 
is a mere matter of strictly mathematical “deformation.” ~The 
stem which we have begun to speak of asa cylinder is not strictly 
so, inasmuch as it tapers off towards its summit. The curve 
which winds evenly around this stem is, accordingly, not a true 
helix, for that term is confined to the curve which winds evenly 
around the cylinder: it is a curve in space which (like the spiral 
curve we have studied in our turbinate shells) partakes of the 
characters of a helix and of a logarithmic spiral, and which is in 
fact a logarithmic spiral with its pole drawn out of its original 
plane by a force acting in the direction of the axis. If we imagine 
a tapering cylinder, or cone, projected, by vertical projection, on 
a plane, it becomes a circular disc; and a helix described about 
the cone necessarily becomes in the disc a logarithmic spiral 
described about a focus which corresponds to the apex of our cone. 
In like manner we may project an identical spiral in space upon 
such surfaces as (for instance) a portion of a sphere or of an ellipsoid ; 
and in all these cases we preserve the spiral configuration, which 
is the more clearly brought into view the more we reduce the 
vertical component by which it was accompanied. The converse 
is, of course, equally true, and equally obvious, namely that any 
logarithmic spiral traced upon a circular disc or spheroidal surface 
will be transformed into a corresponding spiral helix when the 
plane or spheroidal disc is extended into an elongated cone 
approximating to a cylinder. This mathematical conception is 
translated, in botany, into actual fact. The fir-cone may be 
looked upon as a cylindrical axis contracted at both ends, until 


XIv| OR PHYLLOTAXIS 639 


it becomes approximately an ellipsoidal solid of revolution, 
generated about the long axis of the ellipse; and the semi-ellip- 
soidal capitulum of the teasel, the more or less hemispherical one 
of the thistle, and the flattened but still convex one of the sun- 
flower, are all beautiful and successive deformations of what is 
typically a long, conical, and all but cylindrical stem. On the 
other hand, every stem as it grows out into its long cylindrical 
shape is but a deformation of the little spheroidal or ellipsoidal 
surface, or cone, which was its forerunner in the bud. 

This identity of the helical spirals around the stem with spirals 
projected on a plane was clearly recognised by Hofmeister, who 
was accustomed to represent his diagrams of leaf-arrangement 
either in one way or the other, though not in a strictly geometrical 
projection*. 


According to Mr A. H. Church}, who has dealt very carefully 
and elaborately with the whole question of phyllotaxis, the 
logarithmic spirals such as we see in the disc of the sunflower have 
a far greater importance and a far deeper meaning than this brief 
treatment of mine would accord to them: and Sir Theodore Cook, 
in his book on the Curves of Life, has adopted and has helped to 
expound and popularise Mr Church’s investigations. 

Mr Church, regarding the problem as one of “uniform growth,” 
easily arrives at the conclusion that, 7f this growth can be conceived 
as taking place symmetrically about a central point or “pole,” 
the uniform growth would then manifest itself in logarithmic 
spirals, including of course the limiting cases of the circle and 
straight line. With this statement I have little fault to find; it 
is in essence identical with much that I have said in a previous 
chapter. But other statements of Mr Church’s, and many theories 
woven about them by Sir T. Cook and himself, I am less able to 
follow. Mr Church tells us that the essential phenomenon in the 
sunflower disc is a series of orthogonally intersecting logarithmic 
spirals. Unless I wholly misapprehend Mr Church’s meaning, I 
should say that this is very far from essential: The spirals 


* Allgemeine Morphologie der Gewdchse, p. 442, etc. 1868. 
+ Relation of, Phyllotaxis to Mechanical Laws, Oxford, 1901-1903; cf. Ann. 
of Botany, xv, p. 481, 1901. 


640 ON LEAF-ARRANGEMENT [CH. 


intersect isogonally, but orthogonal intersection would be only 
one particular case, and in all probability a very infrequent .one, 
in the intersection of logarithmic spirals developed about a 
common pole. Again on the analogy of the hydrodynamic lines 
of force in certain vortex movements, and of similar lines of 
force in certain magnetic phenomena, Mr Church proceeds to 
argue that the energies of life follow lines comparable to those of 
electric energy, and that the logarithmic spirals of the sunflower 
are, so to speak, lines of equipotential*. And Sir T. Cook 
remarks that this “theory, if correct, would be fundamental for 
all forms of growth, though it would be more easily observed in 
plant construction than in animals.” The parallel I am not able 
to follow. 

Mr Church sees in phyllotaxis an organic mystery, a something 
for which we are unable to suggest any precise cause: a phenomenon 
which is to be referred, somehow, to waves of growth emanating 
from a centre, but on the other hand not to be explained by the 
division of an apical cell, or any other histological factor. As 
Sir T. Cook puts it, “at the growing poimt of a plant where the 
new members are being formed, there is simply nothing to see.” 

But it is impossible to deal satisfactorily, in brief space, either’ 
with Mr Church’s theories, or my own objections to themt. Let 
it suffice to say that I, for my part, see no subtle mystery in the 
matter, other than what lies in the steady production of similar 
growing parts, similarly situated, at similar successive intervals 
of time. If such be the case, then we are bound to have in 

* “The proposition is that the genetic spiral is a logarithmic spiral, homologous 
with the line of current-flow in a spiral vortex; and that in such a system the 
action of orthogonal forces will be mapped out by other orthogonally intersecting 
logarithmic spirals—the ‘parastichies’”; Church, op. cit. 1, p. 42. 

+ Mr Church’s whole theory, if it be not based upon, is interwoven with, Sachs’s 
theory of the orthogonal intersection of cell-walls, and the elaborate theories of 
the symmetry of a growing point or apical cell which are connected therewith. 
According to Mr Church, “the law of the orthogonal intersection of cell-walls at 
a growing apex may be taken as generally accepted” (p. 32): but I have taken a 
very different view of Sachs’s law, in the eighth chapter of the present book. 
With regard to his own and Sachs’s hypotheses, Mr Church makes the following 
curious remark (p. 42): ‘‘ Nor are the hypotheses here put forward more imaginative 
than that of the paraboloid apex of Sachs which remains incapable of proof, or his 


construction for the apical cell of Pteris which does not satisfy the evidence of his 
own drawings.” 


xIv] OR PHYLLOTAXIS 641 


consequence a series of symmetrical patterns, whose nature will 
depend upon the form of the entire surface. If the surface be 
that of a cylinder we shall have a system, or systems, of spiral 
helices: if it be a plane, with an infinitely distant focus, such as 
we obtain by “unwrapping” our cylindrical surface, we shall 
have straight lines; if it be a plane containing the focus within 
itself, or if it be any other symmetrical surface containing the 
focus, then we shall have a system of logarithmic spirals. The 
appearance of these spirals is sometimes spoken of as a “subjective ” 
phenomenon, but the description is inaccurate: it is a purely 
mathematical phenomenon, an inseparable secondary result of 
other arrangements which we, for the time being, regard as primary. 
When the bricklayer builds a factory chimney, he lays his bricks 
in a certain steady, orderly way, with no thought of the spiral 
patterns to which this orderly sequence inevitably leads, and which 
spiral patterns are by no means “subjective”? The designer of 
a wall-paper not only has no intention of producing a pattern 
of criss-cross lines, but on the contrary he does his best to avoid 
them; nevertheless, so long as his design is a symmetrical one, 
the criss-cross intersections inevitably come. 

Let us, however, leave this discussion, and return to the facts 
of the case. 


Our second question, which relates to the numerical coincidences 
so familiar to all students of phyllotaxis, is not to be set and 
answered in a word. 

Let us, for simplicity’s sake, avoid consideration of simultaneous 
or whorled leaf origins, and consider only the more frequent 
cases where a single “genetic spiral’ can be traced throughout 
the entire system. 

It is seldom that this primary, genetic spiral catches the eye, 
for the leaves which immediately succeed one another in this 
genetic order are usually far apart on the circumference of the 
stem, and it is only in close-packed arrangements that the eye 
readily apprehends the continuous series. Accordingly in such 
a case as a fir-cone, for instance, it is certain of the secondary 
Spirals or “parastichies” which catch the eye; and among 
fir-cones, we can easily count these, and we find them to be 


T. G. 4] 


642 ON LEAF-ARRANGEMENT [cH. 


on the whole very constant in number, according to the 
species. 

Thus in many cones, such as those of the Norway spruce, we 
can trace five rows of scales winding steeply up the cone in one 
direction, and three rows winding less steeply the other way; in 
certain other species, such as the common larch, the normal 
number is eight rows in the one direction and five in the other; 
while in the American larch we have again three in the one direction 
and five in the other. It not seldom happens that two arrange- 
ments grade into one another on different parts of one and the 
Same cone. Among other cases in which such spiral series are 
readily visible we have, for instance, the crowded leaves of the 
stone-crops and mesembryanthemums, and (as we have said) the 
crowded florets of the composites. Among these we may find 
plenty of examples in which the numbers of the serial rows are 
similar to those of the fir-cones; but in some cases, as in the daisy 
and others of the smaller composites, we shall be able to trace 
thirteen rows in one direction and twenty-one in the other, or 


perhaps twenty-one and thirty-four; while in a great big sunflower 


we may find (in one and the same species) thirty-four and fifty-five, 
fifty-five and eighty-nine, or even as many as eighty-nine and 
one hundred and forty-four. On the other hand, in an ordinary 
*pentamerous” flower, such as a ranunculus, we may be able to 
trace, in the arrangement of its sepals, petals and stamens, shorter 
spiral series, three in one direction and two in the other. It will 
be at once observed that these arrangements manifest themselves 
in connection with very different things, in the orderly interspacing 
of single leaves and of entire florets, and among all kinds of leaf-like 
structures, foliage-leaves, bracts, cone-scales, and the various 
parts or members of the flower. Again we must be careful to 
note that, while the above numerical characters are by much the 
most common, so much so as to be deemed “normal,” many 
other combinations are known to occur. 

The arrangement, as we have seen, is apt to vary when the 
entire structure varies greatly in size, as in the disc of the sun- 
flower. It is also subject to less regular variation within one and 
the same species, as can always be discovered when we examine 
a sufficiently large sample of fir-cones. For instance, out of 505 


XIV | OR PHYLLOTAXIS 643 


cones of the Norway spruce, Beal* found 92 per cent. in which 
the spirals were in five and eight rows; in 6 per cent. the rows 
were four and seven, and in 4 per cent. they were four and six. 
In each case they were nearly equally divided as regards direction ; 
for instance of the 467 cones shewing the five-eight arrangement, 
the five-series ran in right-handed spirals in 224 cases, and in 
left-handed spirals in 243. 

Omitting the “abnormal” cases, such as we have seen to occur 
in a small percentage of our cones of the spruce, the arrangements 
which we have just mentioned may be set forth as follows, (the 
fractional number used being simply an abbreviated symbol for 
the number of associated helices or parastichies which we can 
count running in the opposite directions): 2/3, 3/5, 5/8, 8/13, 
13/21, 21/34, 34/55, 55/89, 89/144. Now these numbers form a 
very interesting series, which happens to have a number of curious 
mathematical properties+. We see, for instance, that the denomi- 
nator of each fraction is the numerator of the next; and further, 
that each successive numerator, or denominator, is the sum of 
the preceding two. Our immediate problem, then, is to determine, 
if possible, how these numerical coincidences come about, and 
why these particular numbers should be so commonly met with 


* Amer. Naturalist, vu, p. 449, 1873. 
1 


+ This celebrated series, which appears in the continued fraction Ley D cet: 
: 1+ 

and is closely connected with the Sectio awrea or Golden Mean, is commonly called 
the Fibonacci series, after a very learned twelfth century arithmetician (known also 
as Leonardo of Pisa), who has some claims to be considered the introducer of 
Arabic numerals into christian Europe. It is called Lami’s series by some, after 
Father Bernard Lami, a contemporary of Newton’s, and one of the co-discoverers 
of the parallelogram of forces. It was well-known to Kepler, who, in his paper 
De nive sexangula (ct. supra, p. 480), discussed it in connection with the form of 
the dodecahedron and icosahedron, and with the ternary or quinary symmetry of 
the flower. (Cf. Ludwig, F., Kepler iiber das Vorkommen der Fibonaccireihe im 
Pflanzenreich, Bot. Centralbl. Lxvmm, p. 7, 1896). Professor William Allman, 
Professor of Botany in Dublin (father of the historian of Greek geometry), 
speculating on the same facts, put forward the curious suggestion that the cellular 
tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra, 
and that of the monocotyledons or endogens of icosahedra (On the mathematical 
connexion between the parts of Vegetables: abstract of a Memoir read before the 
Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, 
Organogénie végétale, 1, p. 534). 


41—2 


644 ON LEAF-ARRANGEMENT [CH. 


as to be considered “normal” and characteristic features of the 
general phenomenon of phyllotaxis. The following account is 
based on a short paper by Professor P. G. Tait*. 

Of the two following diagrams, Fig. 325 represents the general 
case, and Fig. 326 a particular one, 
for the sake of possibly greater 
simplicity. Both diagrams re- 
present a portion of a branch, or 
Q fir-cone, regarded as cylindrical, 
and unwrapped to form a plane 
surface. A, a, at the two ends 
of the base-line, represent the 

Fig. 325. same initial leaf or scale: Ois a 
leaf which can be reached from 
A by m steps in a right-hand spiral (developed into the straight 
line AO), and by n steps from a in a left-handed spiral a0. Now 
it is obvious in our fir-cone, that we can include all the scales 
upon the cone by taking so many spirals in the one direction, 
and again include them all by so many in the other. Accordingly, 
in our diagrammatic construction, the spirals AO and aO must, 
and always can, be so taken that m spirals parallel to aO, and n 
spirals parallel to AO, shall separately include all the leaves upon 
the stem or cone. 

If m and n have a common factor, J, it can easily be shewn that 
the arrangement is composite, and that there are / fundamental, - 
or genetic spirals, and / leaves (including A) which are situated 
exactly on the line da. That is to say, we have here a whorled 
arrangement, which we have agreed to leave unconsidered in 
favour of the simpler case. We restrict ourselves, accordingly, 
to the cases where there is but one genetic spiral, and when 
therefore m and n are prime to one another. 

Our fundamental, or genetic, spiral, as we have seen, is that 
which passes from A (or «) to the leaf which is situated nearest to 
the base-line da. The fundamental spiral will thus be right- 
handed (A, P, etc.) if P, which is nearer to A than to a, be this 
leaf—left-handed if it be p. That is to say, we make it a con- 
vention that we shall always, for our fundamental spiral, run 

* Proc. Roy. Soc. Edin. vu, p. 391, 1872. 


O 


XIv] OR PHYLLOTAXIS 645 


round the system, from one leaf to the next, by the shortest 
way. 

Now it is obvious, from the symmetry of the figure (as further 
shewn in Fig. 326), that, besides the spirals running along AO and 
aO, we have a series running from the steps on aO to the steps on 
AO. In other words we can find a leaf (S) upon AO, which, hke 
the leaf O, is reached directly by a spiral series from A and from 
a, such that aS includes n steps, and AS (being part of the old 


Fig. 326. 


spiral line AO) now includes m—n steps. And, since m and n 
are prime to one another (for otherwise the system would have 
been a composite or whorled one), it is evident that we can 
continue this process of convergence until we come down to a 
1, 1 arrangement, that is to say to a leaf which is reached by a 
single step, in opposite directions from A and from a, which leaf 
is therefore the first leaf, next to A, of the fundamental or 
generating spiral. 


646 ON LEAF-ARRANGEMENT (cn. 


If our original lines along AO and aO contain, for instance, 
13 and 8 steps respectively (i.e. m= 13, n = 8), then our next 
series, observable in the same cone, will be 8 and (13 — 8) or 5; 
the next 5 and (8 — 5) or 3; the next 3, 2; and the next 2, 1; 
leading to the ultimate condition of 1, 1. These are the very 
series which we have found to be common, or normal; and so 
far as our investigation has yet gone, it has proved to us that, if 
one of these exists, it entails, ipso facto, the presence of the rest. 

In following down our series, according to the above con- 
struction, we have seen that at every step we have changed 
direction, the longer and the shorter sides of our triangle changing 
places every time. Let us stop for a moment, when we come to 
the 1, 2 series, or AT, aT of Fig. 326. It is obvious that there is 
_nothing to prevent us making a new 1, 3 series if we please, by 
continuing the generating spiral through three leaves, and con- 
necting the leaf so reached directly with our initial one. But in 
the case represented in Fig. 326, it is obvious that these two 
series (4, 1, 2, 3, etc., and a, 3, 6, etc.) will be running in the same 
direction ; i.e. they will both be right-handed, or both left-handed 
spirals. The simple meaning of this is that the third leaf of the 
generating spiral was distant from our initial leaf by more than the 
circumference of the cylindrical stem; in other words, that there 
were more than two, but less than three leaves in a single turn of 
the fundamental spiral. . 

Less than two there can obviously never be. When there are 
exactly two, we have the simplest of all possible arrangements, 
namely that in which the leaves are placed alternately on opposite 
sides of the stem. When there are more than two, but less than 
three, we have the elementary condition for the production of the 
series which we have been considering, namely 1, 2; 2, 3; 3, 5, 
etc. To put the latter part of this argument in more precise 
language, let us say that: If, in our descending series, we come to 
steps 1 and ¢, where ¢ is determined by the condition that 1 and 
t+ 1 would give spirals both right-handed, or both left-handed ; 
it follows that there are less than ¢ + 1 leaves in a single turn of 
the fundamental spiral. And, determined in this manner, it is 
found in the great majority of cases, in fir-cones and a host of 
other examples of phyllotaxis, that = 2. In other words, in the 


xIv| OR PHYLLOTAXIS 647 


great majority of cases, we have what corresponds to an arrange- 
ment next in order of simplicity to the simplest case of all: next, 
that is to say, to the arrangement which consists of opposite and 
alternate leaves. 

“These simple considerations,” as Tait says, “explain com- 
pletely the so-called mysterious appearance of terms of the 
recurring series 1, 2, 3, 5, 8, 13, etc.* The other natural series, 
usually but misleadingly represented by convergents to an infinitely 
extended continuous fraction, are easily explained, as above, by 
thking t = 3, 4, 5, etc., etc.” Many examples of these latter series 
have been given by Dickson and other writers. 


We have now learned, among other elementary facts, that 
wherever any one system of helical spirals is present, certain 
others invariably and of necessity accompany it, and are definitely 
related to it. In any diagram, such as Fig. 326, in which we 
represent our leaf-arrangement by means of uniform and regularly 
interspaced dots, we can draw one series of spirals after another, 
and one as easily as another. But in our fir-cone, for instance, 
one particular series, or rather two conjugate series, are always 
conspicuous, while the others are sought and found with com- 
parative difficulty. 

The phenomenon is illustrated by Fig. 327, a—d. The ground- 
plan of all these diagrams is identically the same. The generating 
spiral in each case represents a divergence of 3/8, or 135° of 
azimuth; and the points succeed one another at the same succes- 
sional distances parallel to the axis. The rectangular outlines, 
which correspond to the exposed surface of the leaves or cone- 
scales, are of equal area, and of equal number. Nevertheless 
the appearances presented by these diagrams are very different; 
for in one the eye catches a 5/8 arrangement, in another a 3/5; 
and so on, down to an arrangement of 1/1. The mathematical 
side of this very curious phenomenon I have not attempted to 
investigate. But it is quite obvious that, in a system within 


* The necessary existence of these recurring spirals is also proved, in a 
somewhat different way, by Leslie Ellis, On the Theory of Vegetable Spirals, in 
Mathematical and other Writings, 1853, pp. 358-372. 

+ Proc. Roy. Soc. Edin. vu, p. 397, 1872; Trans. Roy. Soc. Edin. xxv, 
p- 505, 1870-71. 


648 ON LEAF-ARRANGEMENT [cH. 


which various spirals are implicitly contained, the conspicuousness 
of one set or another does not depend upon angular divergence. 
It depends on the relative proportions in length and breadth of 
the leaves themselves; or, more strictly speaking, on the ratio of 


Fig. 327. 


the diagonals of the rhomboidal figure by which each leaf-area is 
circumscribed. When, as in the fir-cone, the scales by mutual 
compression conform to these rhomboidal outlines, their inclined 
edges at once guide the eye in the direction of some one particular 
spiral; and we shall not fail to notice that in such cases the usual 


xiv] OR PHYLLOTAXIS 649 


result is to give us arrangements corresponding to the middle 
diagrams in Fig. 327, which are the configurations in which the 
quadrilateral outlines approach most nearly to a rectangular 
form, and give us accordingly the least possible ratio (under the 
given conditions) of sectional boundary-wall to surface area. 

The manner in which one system of spirals may be caused to 
slide, so to speak, into another, has been ingeniously demonstrated 
by Schwendener on a mechanical model, consisting essentially 
of a framework which can be opened or closed to correspond 
with one after another of the above series of diagrams*. 

The determination of the precise angle of divergence of two 
consecutive leaves of the generating spiral does not enter into the 
above general investigation (though Tait gives, in the same paper, 
a method by which it may be easily determined); and the very fact 
that it does not so enter shews it to be essentially unimportant. 
The determination of so-called “orthostichies,’ or precisely 
vertical successions of leaves, is also unimportant. We have no 
means, other than observation, of determining that one leaf is 
vertically above another, and spiral series such as we have been 
dealing with will appear, whether such orthostichies exist, whether 
they be near or remote, or whether the angle of divergence be 
such that no precise vertical superposition ever occurs. And 
lastly, the fact that the successional numbers, expressed as 
fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final 
term is equal to 0-61803..., the sectio aurea or “golden mean” of 
unity, is seen to be a mathematical coincidence, devoid of 
biological significance; it is but a particular case of Lagrange’s 
theorem that the roots of every numerical equation of the second 
degree can be expressed by a periodic continued fraction. The 
same number has a multitude of curious arithmetical properties. 
It is the final term of all similar series to that with which we have 
been dealing, such for instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 
5/9, etc. It is a number beloved of the circle-squarer, and of all 
those who seek to find, and then to penetrate, the secrets of the 
Great Pyramid. It is deep-set in Pythagorean as well as in 
Euclidean geometry. It enters (as the chord of an angle of 36°), 


* A common form of pail-shaped waste-paper basket, with wide rhomboidal 
meshes of cane, is well-nigh as good a model as is required. 


650 ON LEAF-ARRANGEMENT [cH. 


into the thrice-isosceles triangle of which we have spoken on 
p. 511; it is a number which becomes (by the addition of unity) 
its own reciprocal; its properties never end. To Kepler (as 
Naber tells us) it was a symbol of Creation, or Generation. Its 
recent application to biology and art-criticism by Sir Theodore 
Cook and others is not new. Naber’s book, already quoted, is 
full of it. Zeising, in 1854, found in it the key to all morphology, — 
and the same writer, later on*, declared it to dominate both archi- 
tecture and music. But indeed, to use Sir Thomas Browne’s 
words (though it was of another number that he spoke): “To 
enlarge this contemplation into all the mysteries and secrets ac- 
commodable unto this number, were inexcusable Pythagorisme.”’ 
If this number has any serious claim at all to enter into the 
biological question of phyllotaxis, this must depend on the fact, 
first emphasized by Chauncey Wright}, that, if the successive 
leaves of the fundamental spiral be placed at the particular 
azimuth which divides the circle in this “sectio aurea,” then no 
two leaves will ever be superposed; and thus we are said to have 
“the most thorough and rapid distribution of the leaves round the 
stem, each new or higher leaf falling over the angular space 
between the two older ones which are nearest in direction, so as 
to divide it in the same ratio (K), in which the first two or any 
two successive ones divide the circumference. Now 5/8 and all 
successive fractions differ inappreciably from K.” To this view 
there are many simple objections. In the first place, even 5/8, 
or -625, is but a moderately close approximation to the “golden 
mean”; in the second place the arrangements by which a better 
approximation is got, such as 8/13, 13/21, and the very close 
approximations such as 34/55, 55/89, 89/144, etc., are compara- 
tively rare, while the much less close approximations of 3/5 or 
2/3, or even 1/2, are extremely common. Again, the general 
type of argument such as that which asserts that the plant is 
“aiming at” something which we may call an “ideal angle” is 
one that cannot commend itself to a plain student of physical 
science: nor is the hypothesis rendered more acceptable, when 
Sir T. Cook qualifies it by telling us that “all that a plant can do 


* Deutsche Vierteljahrsschrift, p. 261, 1868. 
+ Memoirs of Amer. Acad. 1x, p. 389. 


XIv] OR PHYLLOTAXIS 651 


is to vary, to make blind shots at constructions, or to “mutate: 
as it is now termed; and the most suitable of these constructions 
will in the long run be isolated by the action of Natural Selection.” 
Finally, and this is the most concrete objection of all, the supposed 
isolation of the leaves, or their most complete “distribution to 
the action of the surrounding atmosphere” is manifestly very little 
affected by any conditions which are confined to the angle of 
azimuth. If we could imagine a case in which all the leaves of 
the stem, or all the scales of a fir-cone, were crushed down to one 
and the same level, into a simple ring or whorl of leaves, then 
indeed they would have their most equable distribution under 
the condition of the “ideal angle,” that is to say of the “golden 
mean.” But if it be (so to speak) Nature’s object to set them 
further apart than they actually are, to give them freer exposure 
to the air than they actually have, then it is surely maniftst that 
the simple way to do so is to elongate the axis, and to set the 
leaves further apart, lengthways on the stem. This has at once 
a far more potent effect than any nice manipulation of the “angie 
of divergence.” For it is obvious that in F(¢ . sin 6) we have a 
greater range of variation by altering 0 than by altering d. We 
come then, without more ado, to the conclusion that the “ Fibon- 
acci series,” and its supposed usefulness, and the hypothesis of 
its introduction into plant-structure through natural seiection. 
are all matters which deserve no place in the plain study oi 
botanical phenomena. As Sachs shrewdly recognised years ago, 
all such speculations as these hark back to a school of mystical 
idealism. 


CHAPTER XV 


ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER 
HOLLOW STRUCTURES 


The eggs of birds and all other hard-shelled eggs, such as those 
of the tortoise and the crocodile, are simple solids of revolution ; 
but they differ greatly in form, according to the configuration of 
the plane curve by the revolution of which the egg is, in a mathe- 
matical. sense, generated. Some few eggs, such as those of the 
owl, the penguin, or the tortoise, are spherical or very nearly so; a 
few more, such as the grebe’s, the cormorant’s or the pelican’s, are 
approximately ellipsoidal, with symmetrical or nearly symmetrical 
ends, and somewhat similar are the so-called “cylindrical” eggs 
of the megapodes and the sand-grouse; the great majority, like 
the hen’s egg, are ovoid, a little blunter at one end than the other; 
and some, by an exaggeration of this lack of antero-posterior 
symmetry, are blunt at one end but characteristically pointed at 
the other, as is the case with the eggs of the guillemot and puffin, 
the sandpiper, plover and curlew. It is an obvious but by no 
means negligible fact that the egg, while often pointed, is never 
flattened or discoidal; it is a prolate, but never an oblate, spheroid. 

The careful study and collection of birds’ eggs would seem to 
have begun with the Count de Marsigl*, the same celebrated 
naturalist who first studied the ‘‘ flowers” of the coral, and who 
wrote the Histoire physique de la mer; and the specific form, as 
well as the colour and other attributes of the egg have been 
again and again discussed, and not least by the many dilettanti 
naturalists of the eighteenth century who soon followed in 
Marsigh’s footsteps f. 

* De avibus circa aquas Danubw vagantibus et de ipsarum Nidis (Vol. v of 
the Danubius Pannonico-mysicus). Hagae Com., 1726. 


+ Sir Thomas Browne had a collection of eggs at Norwich, according to Evelyn, 
in 1671. 


CH. xv] ON THE SHAPES OF EGGS, ETC. 653 


We need do no more than mention Aristotle’s belief, doubtless 
old in his time, that the more poimted egg produces the male 
chicken, and the blunter egg the hen; though this theory survived 
into modern times* and perhaps still lingers on. Several natural- 
ists, such as Giinther (1772) and Biihle (1818), have taken the 
trouble to disprove it by experiment. A more modern and more 
generally accepted explanation has been that the form of the egg 
is in direct relation to that of the bird which has to be hatched 
within—a view that would seem to have been first set forth by 
Naumann and Biihle, in their great treatise on eggs}, and adopted 
by Des Murst and many other well-known writers. 

In a treatise by de Lafresnaye§, an elaborate comparison is 
made between the skeleton and the egg of the various birds, to 
shew, for instance, how those birds with a deep-keeled sternum 
laid rounded eggs, which alone could accommodate the form of the 
young. According to this view, that “Nature had foreseen||”’ 
the form adapted to and necessary for the growing embryo, it 
was easy to correlate the owl with its spherical egg, the diver 
with its elliptical one, and in like manner the round egg of the 
tortoise and the elongated one of the crocodile with the shape of 
the creatures which had afterwards to be hatched therein. A few 
writers, such as Thienemann 4], looked at the same facts the other 
way, and asserted that the form of the egg was determined by 
that of the bird by which it was laid, and in whose body it had 
been conformed. 

In more recent times, other theories, based upon the principles 
of Natural Selection, have been current and very generally accepted, 
to account for these diversities of form. The pointed, conical 
egg of the guillemot is generally supposed to be an adaptation, 


* Cf. Lapierre, in Buffon’s Histoire Naturelle, ed. Sonnini, 1800. 

+ Kier der Vogel Deutschlands, 1818-28 (cit. des Murs, p. 36). 

t Traité d’Oologie, 1860. 

§ Lafresnaye, F. de, Comparaison des ceufs des Oiseaux avec leurs squelettes, 
comme seul moyen de reconnaitre la cause de leurs différentes formes, Rev. Zool.. 
1845, pp. 180-187, 239-244. 

|| Cf. Des Murs, p. 67: “Elle devait encore penser au moment ot ce germe 
aurait besoin de l’espace nécessaire & son accroissement, 2 ce moment ou...il devra 
remplir exactement l’intervalle circonscrit par sa fragile prison, etc.” 

4] Thienemann, F. A. L., Syst. Darstellung der Fortpflanzung der Vogel Europas, 
Leipzig, 1825-38. 


654 ON THE SHAPES OF EGGS [CH. 


advantageous to the species in the circumstances under which 
the egg is laid; the pointed egg is less apt than a spherical one to 
roll off the narrow ledge of rock on which this bird is said to lay 
its solitary egg, and the more pointed the egg, so much the fitter 
and likelier is it to survive. The fact that the plover or the 
sandpiper, breeding in very different situations, lay eggs that are 
also conical, elicits another explanation, to the effect that here 
the conical form permits the many large eggs to be packed closely 
under the mother bird*. Whatever truth there be in these apparent 
adaptations to existing circumstances, it is only by a very hasty 
logic that we can accept them as a vera causa, or adequate 
explanation of the facts; and it is obvious that, in the bird’s egg, 
we have an admirable case for the direct investigation of the 
mechanical or physical significance of its form f. 

Of all the many naturalists of the eighteenth and nineteenth 
centuries who wrote on the subject of eggs, one alone (so far as 
I am aware) ascribed the form of the egg to direct mechanical 
causes. Giinthert, in 1772, declared that the more or less rounded 
or pointed form of the egg is a mechanical consequence of the 
pressure of the oviduct at a time when the shell is yet unformed 
or unsolidified; and that accordingly, to explain the round egg of 
the owl or the kingfisher, we have only to admit that the oviduct 
of these birds is somewhat larger than that of most others, or 
less subject to violent contractions. This statement contains, im 
essence, the whole story of the mechanical conformation of the egg. 

Let us consider, very briefly, the conditions to which the egg 
is subject in its passage down the oviduct§. 

(1) The “egg,” as it enters the oviduct, consists of the yolk 
only, enclosed in its vitelline membrane. As it passes down the 
first portion of the oviduct, the white is gradually superadded, 

* Cf. Newton’s Dictionary of Birds, 1893, p. 191; Szielasko, Gestalt der 
Vogeleier, J. f. Ornith. Lu, pp. 273-297, 1905. 

+ Jacob Steiner suggested a Cartesian oval, r+ mr’ =c, as a general formula 

for all-eggs (cf. Fechner, Ber. séchs. Ges., 1849, p. 57); but this formula (which 


fails in such a case as the guillemot), is purely empirical, and has no mechanical 
foundation. 

{ Giinther, F. C., Sammlung von Nestern und Eyern verschiedener Vogel, 
Niirnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of the Oviduct, 
J. Exp. Zool. vi, pp. 339-359, 1909. 

§ The following account is in part reprinted from Nature, June 4, 1908. 


xv| AND OTHER HOLLOW STRUCTURES 655 


and becomes in turn surrounded by the “shell-membrane.” 
About this latter the shell is secreted, rapidly and at a late period ; 
the egg having meanwhile passed on into a wider portion of the 
oviducal tube, called (by loose analogy, as Owen says) the “uterus.” 
Here the egg assumes its permanent form, here it becomes rigid, 
and it is to this portion of the “oviduct” that our argument 
principally refers. 

(2) Both the yolk and the entire egg tend to fill completely 
their respective membranes, and, whether this be due to growth 
or imbibition on the part of the contents or to contraction on the 
part of the surrounding membranes, the resulting tendency is for 
both yolk and egg to be, in the first instance, spherical, unless 
otherwise distorted by external pressure. 

(3) The egg is subject to pressure within the oviduct, which 
is an elastic, muscular tube, along the walls of which pass peri- 
staltic waves of contraction. These muscular contractions may 
be described as the contraction of successive annuli of muscle, 
giving annular (or radial) pressure to successive portions of the 
egg; they drive the egg forward against the frictional resistance 
of the tube, while tending at the same time to distort its form. 
While nothing is known, so far as I am aware, of the muscular 
physiology of the oviduct, it is well known in the case of the 
intestine that the presence of an obstruction leads to the develop- 
ment of violent contractions in its rear, which waves of contraction 
die away, and are scarcely if at all propagated in advance of the 
obstruction. 

(4) It is known by observation that a hen’s egg is always 
laid blunt end foremost. 

(5) It can be shown, at least as a very common rule, that 
those eggs which are most unsymmetrical, or most tapered off 
posteriorly, are also eggs of a large size relatively to the parent 
bird. The guillemot is a notable case in point, and so also are 
the curlews, sandpipers, phaleropes and terns. We may accord- 
ingly presume that the more pointed eggs are those that are large 
relatively to the tube or oviduct through which they have to pass, 
or, in other words, are those which are subject to the greatest 
pressure while being forced along. So general is this relation 
that we may go still further, and presume with great plausibility 


656 ON THE SHAPES OF EGGS [cH. 


in the few exceptional cases (of which the apteryx is the most 
conspicuous) where the egg is relatively large though not markedly 
unsymmetrical, that in these cases the oviduct itself is in all 
probability large (as Giinther had suggested) in proportion to the 
size of the bird. In the case of the common fowl we can trace a 
direct relation between the size and shape of the egg, for the first 
eggs laid by a young pullet are usually smaller, and at the same 
time are much more nearly spherical than the later ones; and, 
moreover, some breeds of fowls lay proportionately smaller eggs 
than others, and on the whole the former eggs tend to be rounder 
than the latter*. 


% 


We may now proceed to inquire more particularly how the form 
of the egg is controlled by the pressures to which it is subjected. 

The egg, just prior to the formation of the shell, is, as we have 
seen, a fluid body, tending to a spherical shape and enclosed within 
a membrane. 

Our problem, then, is: Given a practically incompressible 
fluid, contained in a deformable capsule, which is either (a) entirely 
inextensible, or (b) slightly extensible, and which is placed in a 
long elastic tube the walls of which are radially contractile, to” 
determine the shape under pressure. 

If the capsule be spherical, inextensible, and completely filled 
with the fluid, absolutely no deformation can take place. The 
few eggs that are actually or approximately spherical, such as 
those of the tortoise or the owl, may thus be alternatively explained 
as cases where little or no deforming pressure has been applied 
prior to the solidification of the shell, or else as cases where the 
capsule was so little capable of extension and so completely filled 
as to preclude the possibility of deformation. 

If the capsule be not spherical, but be imextensible, then 
deformation can take place under the external radial compression, 


* In so far as our explanation involves a shaping or moulding of the egg by 
the uterus or “oviduct” (an agency supplemented by the proper tensions of the 
egg), it is curious to note that this is very much the same as that old view of 
Telesius regarding the formation of the embryo (De rerum natura, vi, cc. 4 and 10), 
which he had inherited from Galen, and of which Bacon speaks (Nov. Ory. cap. 50; 
cf. Fliis’s note). Bacon expressly remarks thut ‘“‘Telesius should have been able 
to shew the like formation in the shells of eggs.” This old theory of embryonic. 
modelling survives only in our usage of the term “‘matrix” for a “mould.” 


Xv] AND OTHER HOLLOW STRUCTURES 657 


only provided that the pressure tends to make the shape more 
nearly spherical, and then only on the further supposition that 
the capsule is also not entirely filled as the deformation proceeds. 
In other words, an incompressible fluid contained in an inexten- 
sible envelope cannot be deformed without puckering of the 
envelope taking place. 

Let us next assume, as the conditions by which this result 
may be avoided, (a) that the envelope is to some extent extensible, 
or (6) that the whole structure grows under relatively fixed 
conditions. The two suppositions are practically identical with 
one another in effect. It is obvious that, on the presumption 
that the envelope is only moderately extensible, the whole structure 
can only be distorted to a moderate degree away from the spherical 
or spheroidal form. 

At all points the shape is determined by the law of the 
distribution of radial pressure within the given region of the tube, 
surface friction helping to maintain the egg in position. If 
the egg be under pressure from the oviduct, but without any 
marked component either in a forward or backward direction, 
the egg will be compressed in the middle, and will tend more or 
less to the form of a cylinder with spherical ends. The eggs of 
the grebe, cormorant, or crocodile may be supposed to receive 
their shape in such circumstances. 

When the egg is subject to the peristaltic contraction of the 
oviduct during its formation, then from the nature and direction 
of motion of the peristaltic wave the pressure will be greatest 
somewhere behind the middle of the egg; in other words, the tube 
is converted for the time being into a more conical form, and the 
simple result follows that the anterior end of the egg becomes the 
broader and the posterior end the narrower. 

With a given shape and size of body, equilibrium in the tube - 
may be maintained under greater radial pressure towards one end 
than towards the other. For example, a cylinder having conical 
ends, of semi-angles @ and 6’ respectively, remains in equilibrium, 
apart from friction, if pcos*@ = p’ cos?0’, so that at the more 
tapered end where @ is small p is small. Therefore the whole 
structure might assume such a configuration, or grow under such 
conditions, finally becoming rigid by solidification of the envelope. 


Gis 4? 


658 ON THE SHAPES OF EGGS [cH. 


According to the preceding paragraph, we must assume some 
initial distribution of pressure, some squeeze applied to the 
posterior part of the egg, in order to give it its tapering form. But, 
that form once acquired, the egg may remain in equilibrium both 
as regards form and position within the tube, even after that 
excess of pressure on the posterior part is relieved. Moreover, 
the above equation shews that a normal pressure no greater and 
(within certain limits) actually less acting upon the posterior part 
than on the anterior part of the egg after the shell is formed will 
be sufficient to communicate to it a forward motion. This is an 
important consideration, for it’ shews that the ordinary form of 
an egg, and even the conical form of an extreme case such as the 
guillemot’s, is directly favourable to the movement of the egg 
within the oviduct, blunt end foremost. 

The mathematical statement of the whole case is as follows: 
In our egg, consisting of an extensible membrane filled with an 
incompressible fluid and under external pressure, the equation of 
the envelope is p,-+ T (1/r+ 1/r’) = P, where p, is the normal 
component of external pressure at a point where 7 and 7’ are the 
radii of curvature, 7' is the tension of the envelope, and P the 
internal fluid pressure. This is simply the equation of an elastic 
surface where 7 represents the coefficient of elasticity; in other 
words, a flexible elastic shell has the same mathematical properties 
as our fluid, membrane-covered egg. And this is the identical 
equation which we have already had so frequent occasion to employ 
in our discussion of the forms of cells; save only that in these 
latter we had chiefly to study the tension 7 (1.e. the surface-tension 
of the semi-fluid cell) and had little or nothing to do with the 
factor of external pressure (p,,), which in the case of the egg becomes 
of chief importance. 

The above equation is the equation of eguilibrium, so that it 
must be assumed either that the whole body is at rest or that its 
motion while under pressure is not such as to affect the result. 
Tangential forces, which have been neglected, could modify the 
form by alteration of 7. In our case we must, and may very 
reasonably, assume that any movement of the egg down the 
oviduct during the period when its form is being impressed upon 
it is very slow, being possibly balanced by the advance of the 


Xv] AND OTHER HOLLOW STRUCTURES 659 


peristaltic wave which causes the movement, as well as by 
friction. 

The quantity 7 is the tension of the enclosing capsule—the 
surrounding membrane. If 7 be constant or symmetrical about 
the axis of the body, the body is symmetrical. But the abnormal 
eggs that a hen sometimes lays, cylindrical, annulated, or quite 
irregular, are due to local weakening of the membrane, in other 
words, to asymmetry of 7. Not only asymmetry of 7, but also 
asymmetry of p,, will render the body subject to deformation, 
and this factor, the unknown but regularly varying, largely 
radial, pressure applied by successive annuli of the oviduct, is the 
essential cause of the form, and variations of form, of the egg. 
In fact, in so far as the postulates correspond near enough to 
actualities, the above equation is the equation of all eggs in the 
universe. At least this is so if we generalise it in the form 
Pn t+ T/r+ T'/r = P in recognition of a possible difference between 
the principal tensions. 

In the case of the spherical egg it is obvious that p, is every- 
where equal. The simplest case is where p, = 0, in other words, 
where the egg is so small as practically to escape deforming 
pressure from the tube. . But we may also conceive the tube to 
be so thin-walled and extensible as to press with practically 
equal force upon all parts of the contained sphere. If while our 
egg be in process of conformation the envelope be free at any 
part from external pressure (that is to say, if p, = 0), then it is 
obvious that that part (if of circular section) will be a portion of 
a sphere. This is not unlikely to be the case actually or approxi- 
mately at one or both poles of the egg, and is evidently the case 
over a considerable portion of the anterior end of the plover’s 
egg. 

In the case of the conical egg with spherical ends, as is niore 
or less the case in the plover’s and the guillemot’s, then at either 
end of the egg r and 7” are identical, and they are greater at the 
blunt anterior end than at the other. If we may assume that p,, 
vanishes at the poles of the egg, then it is plain that 7 varies in 
the neighbourhood of these poles, and, further, that the tension 
T is greatest at and near the small end of the egg. It is here, 
in short, that the egg is most likely to be irregularly distorted or 


42—2 


660 ON THE SHAPES OF EGGS [CH. 


even to burst, and it is here that we most commonly find irregu- 
larities of shape in abnormal] eggs. 

If one portion of the envelope were to become practically stiff 
before p ceases to vary, that would be tantamount to a sudden 
variation of 7’, and would introduce asymmetry by the imposition 
of a boundary condition in addition to the above equation. 

Within the egg lies the yolk, and the yolk is invariably spherical 
or very nearly so, whatever be the form of the entire egg. The 
reason is simple, and lies in the fact that the yolk is itself enclosed 
in another membrane, between which and the outer membrane 
hes a fluid the presence of which makes »,, for the inner membrane 
practically constant. The smallness of friction is indicated by 
the well-known fact that the “germinal spot” on the surface of 
the yolk is always found uppermost, however we may place and 
wherever we may open the egg; that is to say, the yolk easily 
rotates within the egg, bringing its lighter pole uppermost. So, 
owing to this lack of friction in the outer fluid, or white, whatever 
shear is produced within the egg will not be easily transmitted 
to the yolk, and, moreover, owing to the same fluidity, the yolk 
will easily recover its normal sphericity after the egg-shell is 
formed and the unequal pressure relieved. 

These, then, are the general principles involved in, and illus- 
trated by, the configuration of an egg; and they take us as far 
as we can safely go without actual quantitative determinations, 
in each particular case, of the forces concerned. 


In certain cases among the invertebrates, we again find 
instances of hard-shelled eggs which have obviously been 
moulded by the oviduct, or so-called “ootype,” in which they 
have lain: and not merely in such a way as to shew the effects 
of peristaltic pressure upon a uniform elastic envelope, but so 
as to impress upon the egg the more or less irregular form 
of the cavity, within which it had been for a time contained 
and compressed. After this fashion Dr Looss* of Cairo has 


* Journal of Tropical Medicine, 15th June, 1911. I leave this paragraph as it 
was written, though it is now once more asserted that the terminal and lateral- 
spined eggs belong to separate and distinct species of Bilharzia (Leiper, Brit. Med. 
Journ., 18th March, 1916, p. 411). 


xv] AND OTHER HOLLOW STRUCTURES 661 


explained the curious form of the egg in Bilharzia (Schistosoma) 
haematobium, a formidable parasitic worm to which is due a disease 
wide-spread in Africa and Arabia, and an especial scourge of the 
Mecca pilgrims. The egg in this worm is provided at one end 
with a little spine; which now and then is found to be placed not 
terminally but laterally or ventrally, and which when so placed 
has been looked upon as the mark of a supposed new species, 
S. Mansoni. As Looss has now shewn, the little spine must be 
explained as having been moulded within a little funnel-shaped 
expansion of the uterus, just where it communicates with the 
common duct leading from the ovary and yolk-gland; by the 
accumulation of eggs in the ootype, the one last formed is crowded 
into a sideways position, and then, where the side-wall of the egg 
bulges in the funnel-shaped orifice of the duct, a little lateral 
“spine” is formed. In another species, S. japonicum, the egg is 
described as bulging into a so-called “calotte,” or bubble-like 
convexity at the end opposite to the spine. This, I think, may, 
with very little doubt, be ascribed to hardening of the egg-shell 
having taken place just at the period when partial relief from 
pressure was being experienced by the egg in the neighbourhood 
of the dilated orifice of the oviduct. 

This case of Bilharzia is not, from our present point of view, a 
very important one, but nevertheless it is interesting. It ascribes 
to a mechanical cause a curious peculiarity of form; it shews, by 
reference to this mechanical principle, that two conditions which 
were very different to the systematic naturalist’s eye, were really 
only two simple mechanical modifications of the same thing; 
and it destroys the chief evidence for the existence of a supposed 
new species of worm, a continued belief in which, among worms 
of such great pathogenic importance, might lead to gravely 
erroneous pathological deductions. 


On the Form of Sea-urchins 


As a corollary to the problem of the bird’s egg, we may consider 
for a moment the forms assumed by the shells of the sea-urchins. 
These latter are commonly divided into two classes, the Regular 
and the Irregular Echinids. The regular sea-urchins, save in 


G2) =) ON THE SHAPES OF EGGS [CH. 


slight details which do not affect our problem, have a complete 
radial symmetry. The axis of the animal’s body is vertical, 
with mouth below and the intestinal outlet above; and around 
this axis the shell is built as a symmetrical system. It follows 
that in horizontal section the shell is everywhere circular, and we 
shall have only to consider its form as seen in vertical section or 
projection. The irregular urchins (very inaccurately so-called) 
have the anal extremity of the body removed from its central, 
dorsal situation; and it follows that they have now a single plane 
of symmetry, about which the organism, shell and all, is bilaterally 
symmetrical. We need not concern ourselves in detail with the 
shapes of their shells, which may be very simply interpreted, by 
the help of radial co-ordinates, as deformations of the circular or 
“regular” type. 

The sea-urchin shell consists of a membrane, stiffened into 
rigidity by calcareous deposits, which constitute a beautiful 
skeleton of separate, neatly fitting “ossicles.” The rigidity of 
the shell is more apparent than real, for the entire structure is, 
in a sluggish way, plastic; inasmuch as each little ossicle is 
capable of growth, and the entire shell grows by increments to 
each and all of these multitudinous elements, whose individual 
growth involves a certain amount of freedom to move relatively 
to one another; in a few cases the ossicles are so little developed 
that the whole shell appears soft and flexible. The viscera of the 
animal occupy but a small part of the space within the shell, the 
cavity being mainly filled by a large quantity of watery fluid, 
whose density must be very near to that of the external sea-water. 

Apart from the fact that the sea-urchin continues to grow, it 
is plain that we have here the same general conditions as in the 
ego-shell, and that the form of the sea-urchin is subject to a similar 
equilibrium of forces. * But there is this important difference, that 
an external muscular pressure (such as the oviduct administers 
during the consolidation of egg-shell), is now lacking. In its 
place we have the steady continuous influence of gravity, and 
there is yet another force which in all probability we require to 
take into consideration. 

While the sea-urchin is alive, an immense number of delicate 
“tube-feet,” with suckers at their tips, pass through minute pores 


xv] AND OF SEA URCHINS 663 


in the shell, and, like so many long cables, moor the animal to 
the ground. They constitute a symmetrical system of forces, 
with one resultant downwards, in the direction of gravity, and 
another outwards in a radial direction; and if we look upon the 
shell as originally spherical, both will tend to depress the sphere 
into a flattened cake. We need not consider the radial component, 
but may treat the case as that of a spherical shell symmetrically 
depressed under the influence of gravity. This is precisely the 
condition which we have to deal with in a drop of liquid lying on 
a plate; the form of which is determined by its own uniform — 
surface-tension, plus gravity, acting against the uniform internal 
hydrostatic pressure. Simple as this system is, the full mathe- 
matical investigation of the form of a drop is not easy, and we 
can scarcely hope that the systematic study of the Echinodermata 
will ever be conducted by methods based on Laplace’s differential 
equation*; but we have no difficulty in seeing that the various 
forms represented in a series of sea-urchin shells are no other than 
those which we may easily and perfectly imitate in drops. 

In the case of the drop of water (or of any other particular 
liquid) the specific surface-tension is always constant, and the 
pressure varies inversely as the radius of curvature; therefore 
the smaller the drop the more nearly is it able to conserve the 
spherical form, and the larger the drop the more does it become 
flattened under gravity. We can represent the phenomenon by 
using india-rubber balls filled with water, of different sizes; the 
little ones will remain very nearly spherical, but the larger will 
fall down “of their own weight,” into the form of more and more 
flattened cakes; and we see the same thing when we let drops of 
heavy oil (such as the orthotoluidene spoken of on p. 219), fall 
through a tall column of water, the little ones remaining round, 
and the big ones getting more and more flattened as they sink. 
In the case of the sea-urchin, the same series of forms may be 
assumed to occur, irrespective of size, through variations in 7, 
the specific tension, or “strength,” of the enveloping shell. 
Accordingly we may study, entirely from this point of view, 
such a series as the following (Fig. 328). In a very few cases, 
such as the fossil Palaeechinus, we have an approximately spherical 

* Cf. Bashforth and Adams, Theoretical Forms of Drops, etc., Cambridge, 1883. 


664 ON THE SHAPES OF EGGS [CH. 


shell, that is to say a shell so strong that the influence of gravity 
becomes negligible as a cause of deformation. The ordinary 
species of Echinus begin to display a pronounced depression, and 
this reaches its maximum in such soft-shelled flexible forms as 
Phormosoma. On the general question I took the opportunity 
of consulting Mr C. R. Darling, who is an acknowledged expert 
in drops, and he at once agreed with me that such forms as are 
represented in Fig. 328 are no other than diagrammatic illustrations 
of various kinds of drops, “most of which can easily be reproduced 


Fig. 328. Diagrammatic vertical outlines of various Sea-urchins: A, Palaeechinus; 
B, Echinus acutus; C, Cidaris; D, D’ Coelopleurus; E, E’ Genicopatagus; F, 
Phormosoma luculenter; G, P. tenuis; H. Asthenosoma; I, Urechinus. 


in outline by the aid of liquids of approximately equal density to 
water, although some of them are fugitive.’ He found a difficulty 
in the case of the outline which represents Asthenosoma, but the 
reason for the anomaly is obvious; the flexible shell has flattened 
down until it has come in contact with the hard skeleton of the 
jaws, or “Aristotle’s lantern,” within, and the curvature of the 
outlne is accordingly disturbed. The elevated, conical shells 
such as those of Urechinus and Coelopleurus evidently call for 
some further explanation; for there is here some cause at work 


xv] AND OF SEA URCHINS 665 


to elevate, rather than to depress the shell. Mr Darling tells me 
that these forms “are nearly identical in shape with globules I 
have frequently obtained, in which, on standing, bubbles of gas 
rose to the summit and pressed the skin upwards, without being 
able to escape.” The same condition may be at work in the 
sea-urchin; but a similar tendency would also be manifested by 
the presence in the upper part of the shell of any accumulation 
of substance lighter than water, such as is actually present in the 
masses of fatty, oily eggs. 


On the Form and Branching of Blood-vessels 


Passing to what may seem a very different subject, we may 
investigate a number of interesting points in connection with the 
form and structure of the blood-vessels, on the same principle 
and by help of the same equations as those we have used, for 
instance, in studying the egg-shell. 

We know that the fluid pressure (P) within the vessel is 
balanced by (1) the tension (7') of the wall, divided by the radius 
of curvature, and (2) the external pressure (p,), normal to the 
wall: according to our formula 

P =p, + T (1/r + 1/r’). 

If we neglect the external pressure, that is to say any support 
which may be given to the vessel by the surrounding tissues, and 
if we deal only with a cylindrical vein or artery, this formula 
becomes simplified to the form P= T7/R. That is to say, under 
constant pressure, the tension varies as the radius. But the 
tension, per unit area of the vessel, depends upon the thickness 
of the wall, that is to say on the amount of membranous and 
especially of muscular tissue of which it is composed. 

Therefore, so long as the pressure is constant, the thickness 
of the wall should vary as the radius, or as the diameter, of the 
blood-vessel. But it is not the case that the pressure is constant, 
for it gradually falls off, by loss through friction, as we pass from 
the large arteries to the small; and accordingly we find that while, 
for a time, the cross-sections of the larger and smaller vessels are 
symmetrical figures, with the wall-thickness proportional to the 
size of the tube, this proportion is gradually lost, and the walls 


666 ON THE FORM AND BRANCHING [CH. 


of the small arteries, and still more of the capillaries, become 
exceedingly thin, and more so than in strict proportion to the 
narrowing of the tube. 


In the case of the heart we have, within each of its cavities, a 
pressure which, at any given moment, is constant over the whole 
wall-area, but the thickness of the wall varies very considerably. 
For instance, in the left ventricle, the apex is by much the thinnest 
portion, as it is also that with the greatest curvature. We may 
assume, therefore (or at least suspect), that the formula, 
t (1/r + 1/r’) =C, holds good; that is to say, that the thickness (¢é) 
of the wall varies inversely as the mean curvature. This may be 
tested experimentally, by dilating a heart with alcohol under a 
known pressure, and then measuring the thickness of the walls 
in various parts after the whole organ has become hardened. 
By this means it is found that, for each of the cavities, the law 
holds good with great accuracy*. Moreover, if we begin by 
dilating the right ventricle and then dilate the left in like manner, 
until the whole heart is equally and symmetrically dilated, we 
find (1) that we have had to use a pressure in the left ventricle 
from six to seven times as great as in the right ventricle, and 
(2) that the thickness of the walls is just in the same proportion 7. 


A great many other problems of a mechanical or hydro- 
dynamical kind arise in connection with the blood-vesselst, and 
while these are chiefly interesting to the physiologist they have 
also their interest for the morphologist in so far as they bear upon 
structure and form. As an example of such mechanical problems 


* Woods, R. H., On a Physical Theorem applied to tense Membranes, Journ. 
of Anat. and Phys. xxvi, pp. 362-371, 1892. A similar investigation of the 
tensions in the uterine wall, and of the varying thickness of its muscles, was 
attempted by Haughton in his Animal Mechanics, pp. 151-158, 1873. 

+ This corresponds with a determination of the normal pressures (in systole) 
by Krohl, as being in the ratio of 1: 6°8. 

t Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss auf die 
Gestaltung des Arteriensystem, Jen. Zeitschr. xu, p. 267, 1878; Roux, Ueber die 
Verzweigungen der Blutgefassen des Menschen, zhid. xm, p. 205, 1878; Ueber die 
Bedeutung der Ablenkung des Arterienstimmen bei der Astaufgabe, ibid. xu, 
p- 301, 1879; Hess, Walter, Eine mechanisch bedingte Gesetzmissigkeit im Bau 
des Blutgefasssystems, A. f. Entw. Mech. xvi, p. 632, 1903; Thoma, R., Ueber die 
Histogenese und Histomechanik des Blutgefdsssystems, 1893. 


xv] OF BLOOD VESSELS 667 


we may take the conditions which determine or help to determine 
the manner of branching of an artery, or the angle at which its 
branches are given off; for, as John Hunter said*, “To keep up a 
circulation sufficient for the part, and no more, Nature has varied 
the angle of the origin of the arteries accordingly.”” The general 
principle is that the form and arrangement of the blood-vessels is 
such that the circulation proceeds with a minimum of effort, and 
with a minimum of wall-surface, the latter condition leading to a 
minimum of friction and being therefore included in the first. 
What, then, should be the angle of branching, such that there 
shall be the least possible loss of energy in the course of the 
circulation? In order to solve this problem in any particular 
case we should obviously require to know (1) how the loss of 
energy depends upon the distance travelled, and (2) how the loss 
of energy varies with the diameter of the vessel. The loss of 
energy is evidently greater in a narrow tube than in a wide one, 
and greater, obviously, in a long journey than a short. If the 
large artery, AB, give off a comparatively 
narrow branch leading to P (such-as CP, 
or DP), the route ACP is evidently 
shorter than ADP, but on the other i 
hand, by the latter path, the blood has A 
tarried longer in the wide vessel AB, 

and has had a shorter course in the Cf D0’ 
narrow branch. The relative advantage 

of the two paths will depend on the loss 

of energy in the portion CD, as com- A 
pared with that in the alternative portion 

CD’, the latter being short and narrow, the former long and wide. 
If we ask, then, which factor is the more important, length or 
width, we may safely take it that the question is one of degree: 
and that the factor of width will become much the more important 
wherever the artery and its branch are markedly unequal in size. 
In other words, it would seem that for small branches a large 
angle of bifurcation, and for large branches a small one, is always 
the better. Roux has laid down certain rules in regard to the 
branching of arteries, which correspond with the general con- 

* Essays, etc., edited by Owen, 1, p. 134, 1861. 


B 


Fig. 329. 


668 ON THE FORM AND BRANCHING [CH. 


clusions which we have just arrived at. The most important of 
these are as follows: (1) If an artery bifurcate into two equal 
branches, these branches come off at equal angles to the main 
stem. (2) If one of the two branches be smaller than the other, 
then the main branch, or continuation of the original artery, 
makes with the latter a smaller angle than does the smaller or 
“lateral” branch. And (3) all branches which are so small that 
they scarcely seem to weaken or diminish the main stem come off 
from it at a large angle, from about 70° to 90°. 

We may follow Hess in a further investigation of this pheno- 
menon. Let AB be an artery, from which a branch has to be 
given off so as to reach P, and let ACP, ADP, etc., be alternative 
courses which the branch may follow: 
CD, DE, etc., in the diagram, being 
equal distances (=1) along AB. Let 
us call the angles PCD, PCE, 2, 7, 
etc.: and the distances CD’, DE’, by 
which each branch exceeds the next in 
length, we shall call 7,, /,, ete. Now it 
is evident that, of the courses shewn, 
ACP is the shortest which the blood 
can take, but it is also that by which 
its transit through the narrow branch 
is the longest. We may reduce its 
transit through the narrow branch more 
and more, till we come to CGP, or 
rather to a point where the branch 
comes off at right angles to the main 

Fig. 330. stem; but in so doing we very con- 

siderably increase the whole distance 

travelled. We may take it that there will be some intermediate 
point which will strike the balance of advantage. 

Now it is easy to shew that if, in Fig. 330, the route ADP and 
AEP (two contiguous routes) be equally favourable, then any 
other route on either side of these, such as ACP or AFP, must 
be less favourable than either. Let ADP and AEP, then, be 
equally favourable; that is to say, let the loss of energy which 
the blood suffers in its passage along these two routes be equal. 


xv] OF BLOOD VESSELS 669 


Then, if we make the distance DE very small, the angles x, and 
“3 are nearly equal, and may be so treated. And again, if DH 
be very small, then DH’E becomes a right angle, and Jl, (or 
DE) =T cosa. 

But if LZ be the loss of energy per unit distance in the wide 
tube AB, and L’ be the corresponding loss of energy in the narrow 
tube DP, etc., then /Z = 1,L’, because, as we have assumed, the 
loss of energy on the route DP is equal to that on the whole 
route DEP. Therefore IL = IL’ cosz,, and cos z, = L/L’. That 
is to say, the most favourable angle of branching will be such 
that the cosine of the angle is equal to the ratio of the loss of 
energy which the blood undergoes, per unit of length, in the main 
vessel, as compared with that which it undergoes in the branch. 

While these statements are so far true, and while they 
undoubtedly cover a great number of observed facts, yet it is 
plain that, as in all such cases, we must regard them not as a 
complete explanation, but as factors in a complicated phenomenon : 
not forgetting that (as the most learned of all students of the 
heart and arteries, Dr Thomas Young, said in his Croonian 
lecture*) all such questions as these, and all matters connected 
with the muscular and elastic powers of the blood-vessels, 
“belong to the most refined departments of hydraulics.” Some 
other explanation must be sought in order to account for a 
phenomenon which particularly impressed John Hunter’s mind, 
namely the gradually altering angle at which the successive inter- 
costal arteries are given off from the thoracic aorta: the special 
interest of this case arising from the regularity and symmetry of 
the series, for “there is not another set of arteries in the body 
whose origins are so much the same, whose offices are so much 
the same, whose distances from their origin to the place of use, 
and whose uses [?sizes|+ are so much the same.” 


* On the Functions of the Heart and Arteries. Phil. Trans. 1809, pp. 1-31, 
cf. 1808, pp. 164-186; Collected Works, 1, pp. 511-534, 1855. The same lesson is 
conveyed by all such work as that of Volkmann, E. H. Weber and Poiseuille. 
Cf. Stephen Hales’ Statical Essays, u, Introduction: ‘* Especially considering 
that they [i.e. animal Bodies] are in a manner framed of one continued Maze of 
innumerable Canals, in which Fiuids*are incessantly circulating, some with great 
Force and Rapidity, others with very different Degrees of rebated Velocity: 
Hence, etc.” 

+ “Sizes” is Owen’s editorial emendation, which seems amply justified. 


CHAPTER XVI 


ON FORM AND MECHANICAL EFFICIENCY 


There is a certain large class of morphological problems of 
which we have not yet spoken, and of which we shall be able to 
say but little. Nevertheless they are so important, so full of 
deep theoretical significance, and are so bound up with the general 
question of form and of its determination asa result of growth, 
that an essay on growth and form is bound to take account of 
them, however imperfectly and briefly. The phenomena which 
I have in mind are just those many cases where adaptation, in the 
strictest sense, is obviously present, in the clearly demonstrable 
form of mechanical fitness for the exercise of some particular 
function or action which has become inseparable from the life 
and well-being of the organism. 

When we discuss certain so-called “adaptations” to outward 
circumstance, in the way of form, colour and so forth, we are often 
apt to use illustrations convincing enough to certain minds but 
unsatisfying to others—in other words, incapable of demon- 
stration. With regard to colouration, for instance, it is by colours 
“eryptic,’ “warning,” “signalling,” “mimetic,” and so on*, 
that we prosaically expound, and slavishly profess to justify, the 
vast Aristotelian synthesis that Nature makes all things with a 
purpose and “does nothing in vain.” Only for a moment let us 
glance at some few instances by which the modern teleologist 
accounts for this or that manifestation of colour, and is led on 
and on to beliefs and doctrines to which it becomes more and more 
difficult to subscribe. 


* For a more elaborate classification, into colours cryptic, procryptic, anti- 
cryptic, apatetic, epigamic, sematic, episematic, aposematic, etc., see Poulton’s 
Colours of Animals (Int. Scientific Series, Lxvm), 1890; cf. also Meldola, R., 
Variable Protective Colouring in Insects, P.Z.S. 1873, pp. 153-162, ete. 


XVI] THE INTERPRETATION OF COLOUR 671 


Some dangerous and malignant animals are said (in sober 
earnest) to wear a perpetual war-paint, in order to “remind their 
enemies that they had better leave them alone*.”’ The wasp and 
the hornet, in gallant black and gold, are terrible as an army 
with banners; and the Gila Monster (the poison-lizard of the 
Arizona desert) is splashed with scarlet—its dread and_ black 
complexion stained With heraldry more dismal. But the wasp- 
like livery of the noisy, idle hover-flies and drone-flies is but 
stage armour, and in their tinsel suits the little counterfeit cowardly 
knaves mimic the -fighting crew. 

The jewelled splendour of the peacock and the humming-bird, 
and the less effulgent glory of the lyre-bird and the Argus pheasant, 
are ascribed to the unquestioned prevalence of vanity in the one 
sex and wantonness in the otherf. 

The zebra is striped that it may graze unnoticed on the plain, 
the tiger that it may lurk undiscovered in the jungle; the banded 
Chaetodont and Pomacentrid fishes are further bedizened to the 
hues of the coral-reefs in which they dwellt. The tawny lion is 
yellow as the desert sand; but the leopard wears its dappled hide 
to blend, as it crouches on the branch, with the sun-flecks peeping 
through the leaves. 

The ptarmigan and the snowy owl, the arctic fox and the polar 
bear, are white among the snows; but go he north or go he south, 
the raven (like the jackdaw) is boldly and impudently black. 

The rabbit has his white scut, and sundry antelopes their 
piebald flanks, that one timorous fugitive may hie after another, 
spying the warning signal. The primeval terrier or collie-dog 


* Dendy, Evolutionary Biology. p. 336, 1912. 

+ Delight in beauty is one of the pleasures of the imagination; there is no 
limit to its indulgence, and no end to the results which we may ascribe to its 
exercise. But as for the particular “standard of beauty” which the bird (for 
instance) admires and selects (as Darwin says in the Origin, p. 70, edit. 1884), 
we are very much in the dark, and we run the risk of arguing in a circle: for wellnigh 
all we can safely say is what Addison says (in the 412th Spectator)—that each different 
species “is most affected with the beauties of its own kind....Hine merula in nigro 
se oblectat nigra marito;...hinc noctua tetram Canitiem alarum et glaucos miratur 
ocellos.” 

t Cf. Bridge, T. W., Cambridge Natural History (Fishes), vir, p. 173, 1904; 
also. Frisch, K. v., Ueber farbige gene bei Fische, Zool. Jahrb. (Abt. Allg. Zool.), 
Xxx, pp. 171-280, 1914. 


672 ON FORM AND MECHANICAL EFFICIENCY — [cu. 


had brown spots over his eyes that he might seem awake when he 
was sleeping*: so that an enemy might let the sleeping dog he, 
for the singular reason that he imagined him to be awake. And 
a flock of flamingos, wearing on rosy breast and crimson wings 
a garment of invisibility, fades away into the sky at dawn or 
sunset like a cloud incarnadinef. 

To buttress the theory of natural selection the same instances 
of “adaptation” (and many more) are used, which in an earlier 
but not distant age testified to the wisdom of the Creator and 
revealed to simple piety the high purpose of God. In the words 
of a certain learned theologiant, “The free use of final causes to: 
explain what seems obscure was temptingly easy....Hence the 
finalist was often the man who made a hberal use of the zgnava 
ratio, or lazy argument: when you failed to explain a thing by 
the ordinary process of causality, you could “explain” it by 
reference to some purpose of nature or of its Creator. This method 
lent itself with dangerous facility to the well-meant endeavours 
of the older theologians to expound and emphasise the beneficence 
of the divine purpose.” Mutatis mutandis, the passage carries 
its plain message to the naturalist. 

The fate of such arguments or illustrations is always the same. 
They attract and captivate for awhile; they go to the building 
of a creed, which contemporary orthodoxy defends under its 
severest penalties: but the time comes when they lose their 
fascination, they somehow cease to satisfy and to convince, their 
foundations are discovered to be insecure, and in the end no man 
troubles to controvert them. 

But of a very different order from all such “adaptations” as 
these, are those very perfect adaptations of form which, for 
instance, fit a fish for swimming or a bird for flight. Here we are 

* Nature, L, p. 572; Li, pp. 33, 57, 533, 1894-95. 

+ They are “wonderfully fitted for “vanishment’ against the flushed, rich- 
coloured skies of early morning and evening,...their chief feeding-times”; and 
‘‘look like a real sunset or dawn, repeated on the opposite side of the heavens,— 
either east or west as the case may be”: Thayer, Concealing-coloration in the 
Animal Kingdom, New York, 1909, pp. 154-155. This hypothesis, like the rest, 


is not free from difficulty. Twilight is apt to be short in the homes of the flamingo: 
and moreover. Mr Abel Chapman, who watched them on the Guadalquivir, tells 


us that they feed by day. 
{ Principal Galloway, Philosophy of Religion, p. 344, 1914. 


XVI] THE PROBLEM OF ADAPTATION 673 


far above the region of mere hypothesis, for we have to deal with 
questions of mechanical efficiency where statical and dynamical 
considerations can be applied and established in detail. The 
naval architect learns a great part of his lesson from the investi- 
gation of the stream-lines of a fish; and the mathematical study 
of the stream-lines of a bird, and of the principles underlying the 
areas and curvatures of its wings and tail, has helped to lay the 
very foundations of the modern science of aeronautics. When, 
after attempting to comprehend the exquisite adaptation of the 
swallow or the albatross to the navigation of the air, we try to 
pass beyond the empirical study and contemplation of such 
perfection of mechanical fitness, and to ask how such fitness came 
to be, then indeed we may be excused if we stand wrapt in wonder- 
ment, and if our minds be occupied and even satisfied with the 
conception of a final cause. And yet all the while, with no loss 
of wonderment nor lack of reverence, do we find ourselves con- 
strained to believe that somehow or other, in dynamical principles 
and natural law, there lie hidden the steps and stages of physical 
causation by which the material structure was so shapen to its 
ends*. 

But the problems associated with these phenomena are 
difficult at every stage, even long before we approach to the 
unsolved secrets of causation; and for my part I readily confess 
that I lack the requisite knowledge for even an elementary 
discussion of the form of a fish or of a bird. But in the form of 
a bone we have a problem of the same kind and order, so far 
simplified and particularised that we may to some extent deal 
with it, and may possibly even find, in our partial comprehension 
of it, a partial clue to the principles of causation underlying this 
whole class of problems. 


Before we speak of the form of a bone, let us say a word about, 
the mechanical properties of the material of which it is built, in 


* Cf. Professor Flint, in his Preface to Affleck’s translation of Janet’s Causes 
finales: “ We are, no doubt, still a long way from a mechanical theory of organic 
growth, but it may be said to be the quaesitwm of modern science, and no one 
can say that it is a chimaera.” 

+ Cf. Sir Donald MacAlister, How a Bone is Built, Hngl. Ill. Mag. 1884. 


rea Ge 43 


674 ON FORM AND MECHANICAL EFFICIENCY  [on. 


relation to the strength it has to manifest or the forces it has to 
resist: understanding always that we mean thereby the properties 
of fresh or living bone, with all its organic as well as imorganic 
constituents, for dead, dry bone is a very different thing. In all 
the structures raised by the engineer, in beams, pillars and girders 
of every kind, provision has to be made, somehow or other, for 
strength of two kinds, strength to resist compression or crushing, 
and strength to resist tension or pulling asunder. The evenly 
loaded column is designed with a view to supporting a downward 
pressure, the wire-rope, like the tendon of a muscle, is adapted 
only to resist a tensile stress; but in many or most cases the two 
functions are very closely inter-related and combined. The case 
of a loaded beam is a familar one; though, by the way, we are 
now told that it is by no means so simple as it looks, and indeed 
that “the stresses and strains in this log of timber are so complex 
that the problem has not yet been solved in a manner that reason- 
ably accords with the known strength of the beam as found by 
actual experiment*.”’ However, be that as it may, we know, 
roughly, that when the beam is 

loaded in the middle and supported 

at both ends, it tends to be 

bent into an arc, in which con- 

dition its lower fibres are being 

stretched, or are undergoing a 

tensile stress, while its upper 

Fig. 331. fibres are undergoing compres- 

sion. It follows that in some 

intermediate layer there is a “neutral zone,” where the 
fibres of the wood are subject to no stress of either kind. 
In like manner, a vertical pillar if unevenly loaded (as, for 
instance, the shaft of our thigh-bone normally is) will tend to 
bend, and so to endure compression on its concave, and tensile 
stress upon its convex side. In many cases it is the business of 
the engineer to separate out, as far as possible, the pressure-lines 
from the tension-lines, in order to use separate modes of con- 
struction, or even different materials for each. In a suspension- 


* Professor Claxton Fidler, On Bridge Construction, p. 22 (4th ed.), 1909; cf. 
(int. al.) Love’s Elasticity, p. 20 (Historical Introduction), 2nd ed., 1906. 


XVI] THE STRUCTURE OF BONE 675 


bridge, for instance, a great part of the fabric is subject to tensile 
strain only, and is built throughout of ropes or wires; but the 
massive piers at either end of the bridge carry the weight of the 
whole structure and of its load, and endure all the “compression- 
strains” which are inherent in the system. Very much the 
same is the case in that wonderful arrangement of struts and ties 
which constitute, or complete, the skeleton of an animal. The 
“skeleton,” as we see it in a Museum, is a poor and even a mis- 
leading picture of mechanical efficiency*. From the engineer’s 
point of view, it is a diagram showing all the compression-lines, 
but by no means all of the tension-lines of the construction; it 
shews all the struts, but few of the ties, and perhaps we might 
even say none of the principal ones; it falls all to pieces unless 
we clamp it together, as best we can, in a more or less clumsy and 
immobilised way. But in life, that fabric of struts is surrounded 
and interwoven with a complicated system of ties: ligament and 
membrane, muscle and tendon, run between bone and bone; 
and the beauty and strength of the mechanical construction le 
not in one part or in another, but in the complete fabric which 
all the parts, soft and hard, rigid and flexible, tension-bearing 
and pressure-bearing, make up togetherf. 

However much we may find a tendency, whether in nature or 
art, to separate these two constituent factors of tension and 
compression, we cannot do so completely; and accordingly the 
engineer seeks for a material which shall, as nearly as possible, 
offer equal resistance to both kinds of strain. In the following 
table—I borrow it from Sir Donald MacAlister—-we see approxi- 
mately the relative breaking (or tearing) limit and crushing hmit 
in a few substances. ; 

* In preparing or ‘“‘macerating”’ a skeleton, the naturalist nowadays carries 
on the process till nothing is left but the whitened bones. But the old anatomists, 
whose object was not the study of “comparative” morphology but the wider 
theme of comparative physiology, were wont to macerate by easy stages; and in 
many of their most instructive preparations, the ligaments were intentionally left 
in connection with the bones, and as part of the “skeleton.” 

+ In a few anatomical diagrams, for instance in some of the drawings in 
Schmaltz’s Atlas der Anatomie des Pferdes, we may see the system of “ties” 
diagrammatically inserted in the figure of the skeleton. Cf. Gregory, On the 


principles of Quadrupedal Locomotion, Ann. N. Y. Acad. of Sciences, xxu, p. 289, 
1912. 


43—2 


676 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


Average Strength of Materials (in kg. per sq. mm.). 


Tensile Crushing 

strength strength 
Steel 100 145 
Wrought Iron 40 20 
Cast Iron 12 72 
Wood 4 2 
Bone 9-12 13-16 


At first sight, bone seems weak indeed; but it has the great 
‘and unusual advantage that it is very nearly as good for a tie 
as for a strut, nearly as strong to withstand rupture, or tearing 
asunder, as to resist crushing. We see that wrought-iron is only 
half as strong to withstand the former as the latter; while in 
cast-iron there is a still greater discrepancy the other way, for it 
makes a good strut but a very bad tie indeed. Cast-steel is not 
only actually stronger than any of these, but it also possesses, 
like bone, the two kinds of strength in no very great relative 
disproportion. 

When the engineer constructs an iron or steel girder, to take 
the place of the primitive wooden beam, we know that he takes 
advantage of the elementary principle we have spoken of, and 
saves weight and economises material by leaving out as far as 
possible all the middle portion, all the parts in the neighbourhood 
of the “neutral zone”; and in so doing he reduces his girder to 
an upper and lower “flange,” connected together by a “web,” 
the whole resembling, in cross-section, an I or an ZL. 

But it is obvious that, if the strains in the two flanges are to 
be equal as well as opposite, and if the material be such as cast-iron 
or wrought-iron, one or other flange must be made much thicker 
than the other in order that it may be equally strong; and if at 
times the two flanges have, as it were, to change places, or play 
each other’s parts, then there must be introduced a margin of 
safety by making both flanges thick enough to meet that kind of 
stress in regard to which the material happens to be weakest. 
There is great economy, then, in any material which is, as nearly 
as possible, equally strong in both ways; and so we see that, 
from the engineer’s or contractor’s point of view, bone is a very 
good and suitable material for purposes of construction. 


XVI] THE STRUCTURE OF BONE 677 | 


The I or the H-girder or rail is designed to resist bending in one 
particular direction, but if, as in a tall pillar, it be necessary to 
resist bending in all directions alike, it is obvious that the tubular 
or cylindrical construction best meets the case; for it is plain 
that this hollow tubular pillar is but the I-girder turned round 
every way, in a “solid of revolution,” so that on any two opposite 
sides compression and tension are equally met and resisted, and 
there is now no need for any substance at all in the way of web 
or “filling” within the hollow core of the tube. And it is not only 
in the supporting pillar that such a construction is useful; it is 
appropriate in every case where stiffness is required, where bending 
has to be resisted. The long bone of a bird’s wing has little or 
no weight to carry, but it has to withstand powerful bending 
moments; and in the arm-bone of a long-winged bird, such as 
an albatross, we see the tubular construction manifested in its 
perfection, the bony substance being reduced to a thin, perfectly 
cylindrical, and almost empty shell. The quill of the bird’s 
feather, the hollow shaft of a reed, the thin tube of the wheat- 
straw bearing its heavy burden in the ear, are all illustrations 
which Galileo used in his account of this mechanical principle*. 

Two points, both of considerable importance, present themselves 
here, and we may deal with them before we go further. In the 
first place, it is not difficult to see that, in our bending beam, the 
strain is greatest at its middle; if we press our walking-stick hard 
against the ground, it will tend to snap midway. Hence, if our 
cylindrical column be exposed to strong bending stresses, it will 
be prudent and economical to make its walls thickest in the middle 
and thinning off gradually towards the ends; and if we look at 
a longitudinal section of a thigh-bone, we shall see that this is 
just what nature has done.’ The thickness of the walls is nothing 
less than a diagram, or “graph,” of the “bending-moments ” 
from one point to another along the length of the bone. ° 

The second point requires a little more explanation. Uf we 

* Galileo, Dialogues concerning Two New Sciences (1638), Crew and Salvio’s 
translation, New York, 1914, p. 150; Opere, ed. Favaro, vii, p. 186. Cf. Borelli, 
De Motu Animalium, 1, prop. CLXxx, 1685. Cf. also Camper, P., La structure des 
os dans les oiseaux, Opp. m1, p. 459, ed. 1803; Rauber, A., Galileo itiber Knochen- 


formen, Morphol. Jahrb. vu, pp. 327. 328, 1881; Paolo Enriques, Della economia 
di sostanza nelle osse cave, Arch. f. Ent. Mech. xx, pp. 427-465, 1906. 


678 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


imagine our loaded beam to be supported at one end only (for 
instance, by being built into a wall), so as to form what is called 
a “bracket” or “cantilever,” then we can 
see, without much difficulty, that the lines 
of stress in the beam run somewhat as in 
the accompanying diagram. Immediately 
under the load, the “compression-lines”’ 
tend to run vertically downward; but 
Fig. 332. where the bracket is fastened to the 

wall, there is pressure directed horizon- 

tally against the wall in the lower part of the surface of 
attachment; and the vertical beginning and the horizontal end 
of these pressure-lines must be continued into one another in the 
form of some even mathematical curve—which, as it happens, 
is part of a parabola. The tension-lines are identical in form 
with the compression-lines, of which they constitute the “mirror- 
image’; and where the two systems intercross, they do so at 
right angles, or “orthogonally” to one another. Such systems 
of stress-lines as these we shall deal with again; but let us take 
note here of the important, though well-nigh obvious fact, that 
while in the beam they both unite to carry the load, yet it is 
always possible to weaken one set of lines at the expense of the 
other, and in some cases to do altogether away with one set or 
the other. For example, when we replace our end-supported 
beam by a curved bracket, bent upwards or downwards as the 
case may be, we have evidently cut away in the one case the 
greater part of the tension-lines, and in the other the greater part 
of the compression-lines. And if instead of bridging a stream 
with our beam of wood we bridge it with a rope, it is evident that 
this new construction contains all the tension-lines, but none of 
the compression-lines of the old. The biological interest connected 
with this principle lies chiefly in the mechanical construction of 
the rush or the straw, or any other typically cylindrical stem. 
The material of which the stalk is constructed is very weak to 
withstand compression, but parts of it have a very great tensile 
strength. Schwendener, who was both botanist and engineer, 
has elaborately investigated the factor of strength in the 
cylindrical stem, which Galileo was the first to call attention to. 


> a THE STRUCTURE OF BONE 679 


-Schwendener* shewed that the strength was concentrated in the 
little bundles of “bast-tissue,” but that these bast-fibres had a 
tensile strength per square mm. of section, up to the limit of 
elasticity, not less than that of steel-wire of such quality as was 
in use in his day. ; 

For instance, we see in the following table the load which 
various fibres, and various wires, were found capable of sustaining, 
not up to the breaking-point, but up to the “elastic limit,” or 
point beyond which complete recovery to the original length took 
place no longer after release of the load. 


Stress, or load in gms. Strain. or amount 


per sq. mm., at of stretching, 
Limit of Elasticity per mille 
Secale cereale 15-20 4-4 
Lilium auratum 19 7-6 
Phormium tenax 20 ; 13-0 
Papyrus antiquorum 20 15:2 
Molinia coerulea 22 pe 
Pincenectia recurvata 25 14-5 
Copper wire 12-1 1:0 
Brass 3 13:3 1:35 
Tron 5 21:9 1:0 
Steel 3 24-6 1-2 


In other respects, it is true, the plant-fibres were inferior to 
the wires; for the former broke asunder very soon after the 
limit of elasticity was passed, while the iron-wire could stand, 
before snapping, three times the load which was measured by its 
limit of elasticity: in the language of a modern engineer, the 
bast-fibres had a low “yield-point,” little above the elastic limit. 
But nevertheless, within certain limits, plant-fibre and wire were 
just as good and strong one as the other. And then Schwendener 
proceeds to shew, in many beautiful diagrams, the various ways 
in which these strands of strong tensile tissue are arranged in 
various cases: sometimes, in the simpler cases, forming numerous 
small bundles arranged in a peripheral ring, not quite at the 
periphery, for a certain amount of space has to be left for living 
and active tissue; sometimes in a sparser ring of larger and 


* Das mechanische Prinzip im anatomischen Bau der Monocotylen, Leipzig, 
1874. 


680 ON FORM AND MECHANICAL EFFICIENCY _ [cu. 


stronger bundles; sometimes with these bundles further strength- 
ened by radial balks or ridges; sometimes with all the fibres set 
close together in a continuous hollow 
cylinder. In the case figured in Fig. 
_ 333 Schwendener calculated that the 
resistance to bending was at least 
twenty-five times as great as it would 
have been had the six main bundles 
been brought close together in a solid 
core. In many cases the centre of 
the stem is altogether empty; in all 
other cases it is filled with soft tissue, 
suitable for the ascent of sap or other 
functions, but never such as to confer mechanical rigidity. Ina 
tall conical stem, such as that of a palm-tree, we can see not only 
these principles in the construction of the cylindrical trunk, but 
we can observe, towards the apex, the bundles of fibre curving 
over and intercrossing orthogonally with one another, exactly 
after the fashion of our stress-lines in Fig. 332; but of course, in 
this case, we are still dealing with tensile members, the opposite 
bundles taking on in turn, as the tree sways, the alternate 
function of resisting tensile strain*. 


it 


Iie 


Let us now come, at last, to the mechanical structure of bone, 
of which we find a well-known and classical illustration in the 
various bones of the human leg. In the case of the tibia, the bone 
is somewhat widened out above, and its hollow shaft is capped 
by an almost flattened roof, on which the weight of the body 
directly rest. It is obvious that, under these circumstances, the 
engineer would find it necessary to devise means for supporting 
this flat roof, and for distributing the vertical pressures which 
impinge upon it to the cylindrical walls of the shaft. 


* For further botanical illustrations, see (int. al.) Hegler, Einfluss der Zug- 
kraften auf die Festigkeit und die Ausbildung mechanischer Gewebe in Pflanzen, 
SB. stichs. Ges. d. Wiss. p. 638, 1891; Kny, L,, Einfluss von Zug und Druck auf 
die Richtung der Scheidewande in sich teilenden Pflanzenzellen, Ber. d. bot. 
Gesellsch. xtv, 1896; Sachs, Mechanomorphose und Phylogenie, Flora, LXXVII, 
1894; cf. also Pfliiger, Einwirkung der Schwerkraft, etc., iiber die Richtung der 
Zelltheilung, Archiv, xxxiv, 1884. 


XVI] THE STRUCTURE OF BONE 681 


In the case of the bird’s wing-bone, the hollow of the bone is 
practically empty, as we have already said, being filled only with 
air save for a thin layer of living tissue immediately within the 
cylinder of bone; but in our own bones, and all weight-carrying 
bones in general, the hollow space is filled with marrow, blood- 
vessels and other tissues; and among these living tissues lies a 
fine lattice-work of little interlaced “trabeculae” of bone, forming 


Fig. 334. Head of the human femur in section. (After Schafer, from 
a photo by Prof. A. Robinson.) 


the so-called “cancellous tissue.” The older anatomists were 
content to describe this cancellous tissue as a sort of “spongy 
network,” or irregular honeycomb, until, some fifty years ago, a 
remarkable discovery was made regarding it. It was found by 
Hermann Meyer (and afterwards shewn in greater detail by 
Julius Wolff and others) that the trabeculae, as seen in a longi- 
tudinal section of a long bone, were arranged in a very definite 
and orderly way; in the femur, they spread in beautiful curving 


682 ON FORM AND MECHANICAL EFFICIENCY — [ca. 


lines from the head to the tubular shaft of the bone, and these 
bundles of lines were crossed by others, with so nice a regu- 
larity of arrangement that each intercrossing was as nearly as 
possible an orthogonal one: that is to say, the one set of fibres 
crossed the other everywhere at right angles. A great engineer, 
Professor Culmann of Ziirich (to whom, by the way, we owe the 
whole modern method of “graphic statics”), happened to see 
some of Meyer’s drawings and preparations, and he recognised 
in a moment that in the arrangement of the trabeculae we had 


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nothing more nor less than a diagram of the lines of stress, or 
directions of compression and tension, in the loaded structure: 
in short, that nature was strengthening the bone in precisely the 
manner and direction in which strength was needed. In the 
accompanying diagram of a crane-head, by Culmann, we recognise 
a slight modification (caused entirely by the curved shape of the 
structure) of the still simpler lines of tension and compression 
which we have already seen in our end-supported beam as 
represented in Fig. 332. In the shaft of the crane, the concave 


Xvi] THE STRUCTURE OF BONE 683 


6 


or inner side, overhung by the loaded head, is the “compression- 
member’; the outer side is the “tension-member’”; and the 
pressure-lines, starting from the loaded surface, gather themselves 
together, always in the direction of the resultant pressure, till 
they form a close bundle running down the compressed side 
of the shaft: while the tension-lines, running upwards along the 
opposite side of the shaft, spread out through the head, ortho- 
gonally to, and linking together, the system of compression-lines. 
The head of the femur (Fig. 335) is a little more complicated in 
form and a little less symmetrical than Culmann’s diagrammatic 
crane, from which it chiefly differs in the fact that the load is 
divided into two parts, that namely which is borne by the head 
of the bone, and that smaller portion which rests upon the great 
trochanter; but this merely amounts to saying that a notch has 
been cut out of the curved upper surface of the structure, and we 
have no difficulty in seeing that the anatomical arrangement of 
the trabeculae follows precisely the mechanical distribution of 
compressive and tensile stress or, in other words, accords perfectly 
with the theoretical stress-diagram of the crane. The lines of 
stress are bundled close together along the sides of the shaft, and 
lost or concealed there in the substance of the solid wall of bone; 
but in and near the head of the bone, a peripheral shell of bone 
does not suffice to contain them, and they spread out through the 
central mass in the actual concrete form of bony trabeculae*. 
* Among other works on the mechanical construction of bone see: Bourgery, 
Traité de Vanatomie (I. Ostéologie), 1832 (with admirable illustrations of trabecular 
structure); Fick, L., Die Ursachen der Knochenformen, Gottingen, 1857; Meyer. H., 
Die Architektur der Spongiosa, Archiv f. Anat. und Physiol. xuvu, pp. 615-628, 
1867; Statik uw. Mechanik des menschlichen Knochengeriistes, Leipzig, 1873; 
Wolff, J., Die innere Architektur der Knochen, Arch. f. Anat. und Phys. L, 1870; 
Das Gesetz der Transformation bei Knochen, 1892; von Ebner, V., Der feinere Bau 
der Knochensubstanz, Wiener Bericht, uxxi1, 1875; Rauber, Anton, Elastizitat und 
Festigkeit der Knochen, Leipzig, 1876; O. Meserer, Elast. u. Festigk. d. mensch- 
lichen Knochen, Stuttgart, 1880; MacAlister, Sir Donald, How a Bone is Built, 
English Illustr. Mag. pp. 640-649, 1884; Rasumowsky, Architektonik des Fuss- 
skelets, Int. Monatsschr. f. Anat. p. 197, 1889; Zschokke, Weitere Unters. iiber das 
Verhdltniss der Knochenbildung zur Statik und Mechanik des Vertebratenskelets, 
Ziirich, 1892; Roux, W., Ges. Abhandlungen iiber Entwicklungsmechanik der 
Organismen, Bd. I, Funktionelle Anpassung, Leipzig, 1895; Triepel. H., Die 
Stossfestigkeit der Knochen, Arch. f. Anat. vu. Phys. 1900; Gebhardt, Funktionell 


wichtige Anordnungsweisen der feineren und gréberen Bauelemente des Wirbel- 
thierknochens, ete., Arch. f. Entw. Mech. 1900-1910; Kirchner, A., Architektur 


684 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


Mutatis mutandis, the same phenomenon may be traced in any 
other bone which carries weight and is liable to flexure; and in 
the os calcis and the tibia, and more or less 1n all the bones of the 
lower limb, the arrangement is found to be very simple and 
clear. 

Thus, in the os calcis, the weight resting on the head of the 
bone has to be transmitted partly through the backward-projecting 
heel to the ground, and partly forwards through its articulation 
with the cuboid bone, to the arch of the foot. We thus have, 
very much as in a triangular roof-tree, two compression-members, 
sloping apart from one another; and these have to be bound 


Fig. 336. Diagram of stress-lines in the human foot. (From Sir 
D. MacAlister, after H. Meyer.) 


together by a “tie” or tension-member, corresponding to the 
third, horizontal member of the truss. 


So far, dealing wholly with the stresses and strains due to 
tension and compression, we have altogether omitted to speak 
of a third very important factor in the engineer’s calculations, 
namely what is known as “shearing stress.” A shearing force is 
one which produces “angular distortion” in a figure, or (what 
comes to the same thing) which tends to cause its particles to 
der Metatarsalien, A. f. E. M. xx1v, 1907; Triepel, Herm., Die trajectorielle 
Structuren (in Hinf. in die Physikalische Anatomie, 1908); Dixon, A. F., 


Architecture of the Cancellous Tissue forming the Upper End of the Femur, 
Journ. of Anat. and Phys. (3) xLiv, pp. 223-230, 1910. 


XVI] THE STRUCTURE OF BONE 685 


slide over one another. A shearing stress is a somewhat com- 
plicated thing, and we must try to illustrate it (however 
imperfectly) in the simplest possible way. If we build up a pillar, 
for instance, of a pile of flat horizontal slates, or of a pack of 
cards, a vertical load placed upon it will produce compression, but 
will have no tendency to cause one card to slide, or shear, upon 
another; and in like manner, if we make up a cable of parallel 
wires and, letting it hang vertically, load it evenly with a weight, 
again the tensile stress produced has no tendency to cause one 
wire to slip or shear upon another. But the case would have 


Fig. 337. Trabecular structure of the os calcis. (From MacAlister.) 


been very different if we had built up our pillar of cards or slates 
lying obhquely to the lines of pressure, for then at once there 
would have been a tendency for the elements of the pile to slip 
and slide asunder, and to produce what the geologists call “a 
fault” in the structure. 


Somewhat more generally, if AB be a bar, or pillar, of cross-section a 
under a direct load P, giving a stress per unit area = p, then the whole 
pressure P= pa. Let CD be an oblique section, inclined at an angle 6 to the 
cross-section; the pressure on CD will evidently be = pa cos 6. But at any 
point O in CD, the pressure P may be resolved into the force Q acting along 
CD, and N perpendicular to it: where N = P cos 6, and Q= P sin 6 = pasin 6. 
The whole force Q@ upon CD=q.area of CD, which is = q.a/cos @. 


686 ON FORM AND MECHANICAL EFFICIENCY [cx. 


Therefore ga/cos 6 = pasin 6, therefore gq = psin 6 cos 6, = }p sin 26. 
Therefore when sin 26 = 1, that is, when @ = 45°, q is a maximum, and 
=p/2; and when sin 26=0, that is when 6=0° 

Vv v or 90°, then qg vanishes altogether. 


This is as much as to say, that a 

D shearing stress vanishes altogether along 

the lines of maximum compression or 

tension; it has a definite value in all 

other positions, and a maximum value 

when it is inclined at 45° to either, or 

C half-way between the two. This may be 

further illustrated in various simple ways. 

nH» A B When we submit a cubical block of iron 

teen to compression in the testing machine, it 

Z does not tend to give way by crumbling 

all to pieces; but as a rule it disrupts by shearing, and along 

some plane approximately at 45° to the axis of compression. 

Again, in the beam which we have already considered under a 

bending moment, we know that if we substitute for it a pack of 

cards, they will be strongly sheared on one another; and the 

shearing stress is greatest in the “neutral zone,” where neither 

tension nor compression is manifested: that is to say in the line 

which cuts at equal angles of 45° the orthogonally intersecting 
lines of pressure and tension. 

In short we see that, while shearing stresses can by no means 
be got rid of, the danger of rupture or breaking-down under 
shearing stress is completely got rid of when we arrange the 
materials of our construction wholly along the pressure-lines and 
tension-lines of the system; for along these lines there is no shear. 

To apply these principles to the growth and development of 
our bone, we have only to imagine a little trabecula (or group of 
trabeculae) being secreted and laid down fortuitously in any 
direction within the substance of the bone. If it lie in the 
direction of one of the pressure-lines, for instance, it will be in 
a position of comparative equilibrium, or minimal disturbance ; 
but if it be inclined obliquely to the pressure-lines, the shearing 
force will at once tend to act upon it and move it away. This 
is neither more nor less than what happens when we comb our 


XVI] THE STRUCTURE OF BONE 687 


hair, or card a lock of wool: filaments lying in the direction of 
the comb’s path remain where they were; but the others, under 
the influence of an oblique component of pressure, are sheared 
out of their places till they too come into coincidence with the 
lines of force. So straws show how the wind blows—or rather 
how it has been blowing. For every straw that lies askew to the 
wind’s path tends to be sheared into it; but as soon as it has 
come to lie the way of the wind it tends to be disturbed no 
more, save (of course) by a violence such as to hurl it bodily 
away. 

In the biological aspect of the case, we must always re- 
member that our bone is not only a living, but a highly plastic 
structure; the little trabeculae are constantly being formed and 
deformed, demolished and formed anew. Here, for once, it is 
safe to say that “heredity” need not and cannot be invoked to 
account for the configuration and arrangement of the trabeculae: 
for we can see them, at any time of life, in the making, under the 
direct action and control of the forces to which the system is 
exposed. If a bone be broken and so repaired that its parts lie 
somewhat out of their former place, so that the pressure- and 
tension-lines have now a new distribution, before many weeks are 
over the trabecular system will be found to have been entirely 
remodelled, so as to fall into line with the new system of forces. 
And as Wolff pointed out, this process of reconstruction extends 
a long way off from the seat of injury, and so cannot be looked 
upon as a mere accident of the physiological process of healing 
and repair; for instance, it may happen that, after a fracture of 
the shaft of a long bone, the trabecular meshwork is wholly altered 
and reconstructed within the distant extrenuties of the bone. 
Moreover, in cases of transplantation of bone, for example when 
a diseased metacarpal is repaired by means of a portion taken 
from the lower end of the ulna, with astonishing quickness the 
plastic capabilities of the bony tissue are so manifested that 
neither in outward form nor inward structure can the old portion 
be distinguished from the new. 

Herein then lies, so far as we can discern it, a great part at 
least of the physical causation of what at first sight strikes us as 
a purely functional adaptation: as a phenomenon, in other words, 


688 ON FORM AND MECHANICAL EFFICIENCY  [cz. 


whose physical cause is as obscure as its final cause or anit is, 
apparently, manifest. 


Partly associated with the same phenomenon, and partly to 
be looked upon (meanwhile at least) as a fact apart, is the very 
important physiological truth that a condition of strain, the 
result of a stress, is a direct stimulus to growth itself. This indeed 
is no less than one of the cardinal facts of theoretical biology. 
The soles of our boots wear thin, but the soles of our feet grow 
thick, the more we walk upon them: for it would seem that the 
living cells are “stimulated” by pressure, or by what we call 
“exercise,” to increase and multiply. The surgeon knows, when 
he bandages a broken limb, that his bandage is doing something 
more than merely keeping the parts together: and that the even, 
constant pressure which he skilfully applies is a direct encourage- 
ment of growth and an active agent in the process of repair. In the 
classical experiments of Sédillot*, the greater part of the shaft of the 
tibia was excised in some young puppies, leaving the whole weight 
of the body to rest upon the fibula. The latter bone is normally 
about one-fifth or sixth of the diameter of the tibia; but under 
the new conditions, and under the “stimulus” of the increased 
load, it grew till it was as thick or even thicker than the normal 
bulk of the larger bone. Among plant tissues this phenomenon 
is very apparent, and in a somewhat remarkable way; for a strain 
caused by a constant or increasing weight (such as that in the 
stalk of a pear while the pear is growing and ripening) produces 
a very marked increase of strength without any necessary increase 
of bulk, but rather by some histological, or molecular, alteration 
of the tissues. Hegler, and also Pfeffer, have investigated this 
subject, by loading the young shoot of a plant nearly to its 
breaking point, and then redetermining the breaking-strength 
after a few days. Some young shoots of the sunflower were found 
to break with a strain of 160 gms. ; but when loaded with 150 gms., 
and retested after two days, they were able to support 250 gms. ; 
and being again loaded with something short of this, by next day 
they sustained 300 gms., and a few days later even 400 gms. 


* Sédillot, De l’influence des fonctions sur la structure et la forme des organes; 
O. R. 11x, p. 539, 1864; cf. Lx, p. 97, 1865, Lxvut, p. 1444. 1869. 


xvi] ON STRESS AND STRAIN 689 


Such experiments have been amply confirmed, but so far as 
I am aware, we do not know much more about the matter: we 
do not know, for instance, how far the change is accompanied by 
increase in number of the bast-fibres, through transformation of 
other tissues; or how far it is due to increase in size of these 
fibres; or whether it be not simply due to strengthening of the 
original fibres by some molecular change. But I should be much 
inclined to suspect that the latter had a good deal to do with the 
phenomenon. We know nowadays that a railway axle, or any 
other piece of steel, is weakened by a constant succession of 
frequently interrupted strains; it is said to be “fatigued,” and 
its strength is restored by a period of rest. The converse effect 
of continued strain in a uniform direction may be illustrated by 
a homely example. The confectioner takes a mass of boiled 
. sugar or treacle (in a particular molecular condition determined 
by the temperature to which it has been exposed), and draws the 
soft sticky mass out into a rope; and then, folding it up lengthways, 
he repeats the process again and again. At first the rope is pulled 
out of the ductile mass without difficulty; but as the work goes 
on it gets harder to do, until all the man’s force is used to stretch 
the rope. Here we have the phenomenon of increasing strength, 
following mechanically on a rearrangement of molecules, as the 
original isotropic condition is transmuted more and more into 
molecular asymmetry or anisotropy; and the rope apparently 
“adapts itself” to the increased strain which it is called on to bear, 
all after a fashion which at least suggests a parallel to the increasing 
strength of the stretched and weighted fibre in the plant. For 
increase of strength by rearrangement of the particles we have 
already a rough illustration in our lock of wool or hank of tow. 
The piece of tow will carry but little weight while its fibres are 
tangled and awry: but as soon as we have carded it out, and 
brought all its long fibres parallel and side by side, we may at once 
make of it a strong and useful cord. 

In some such ways as these, then, it would seem that we may 
co-ordinate, or hope to co-ordinate, the phenomenon of growth 
with certain of the beautiful structural phenomena which present 
themselves to our eyes as “ provisions,” or mechanical adaptations, 
for the display of strength where strength is most required. 


™ G: 44 


690 ON FORM AND MECHANICAL. EFFICIENCY _ [cu. 


That is to say, the origin, or causation, of the phenomenon would 
seem to lie, partly in the tendency of growth to be accelerated 
under strain: and partly in the automatic effect of shearing 
strain, by which it tends to displace parts which grow obliquely 
to the direct lines of tension and of pressure, while leaving those 
in place which happen to lie parallel or perpendicular to those 
lines: an automatic effect which we can probably trace as working 
on all scales of magnitude, and as accounting therefore for the 
rearrangement of minute particles in the metal or the fibre, as 
well as for the bringing into line of the fibres themselves within 
the plant, or of the little trabeculae within the bone. 


But we may now attempt to pass from the study of the 
individual bone to the much wider and not less beautiful problems 
of mechanical construction which are presented to us by the 
skeleton as a whole. Certain problems of this class are by no 
means neglected by writers on anatomy, and many have been 
handed down from Borelli, and even from older writers. For 
instance, it is an old tradition of anatomical teaching to point 
out in the human body examples of the three orders of levers*; 
again, the principle that the limb-bones tend to be shortened in 
order to support the weight of a very heavy animal is well under- 
stood by comparative anatomists, in accordance with Huler’s law, 
that the weight which a column liable to flexure is capable of 
supporting varies inversely as the square of its length; and again, 
the statical equilibrium of the body, in relation for imstance to 
the erect posture of man, has long been a favourite theme of the 
philosophical anatomist. But the general method, based upon — 
that of graphic statics, to which we have been introduced in our 
study of a bone, has not, so far as I know, been applied to the 
general fabric of the skeleton. Yet itis plain that each bone plays 


* H.g. (1) the head, nodding backwards and forwards on a fulcrum, represented 
by the atlas vertebra, lying between the weight and the power; (2) the foot, raising 
on tip-toe the weight of the body against the fulerum of the ground, where the 
weight is between the fulcrum and the power, the latter being represented by the 
tendo Achillis; (3) the arm, lifting a weight in the hand, with the power (i.e. the 
biceps muscle) between the fulcrum and the weight. (The second case, by the way, 
has been much disputed; cf. Haycraft in Schafer’s Textbook of Physiology, p. 251. 
1900.) 


xvi] THE COMPARATIVE ANATOMY OF BRIDGES 691 


a part in relation to the whole body, analogous to that which a 
little trabecula, or a little group of trabeculae, plays within the 
bone itself: that is to say, in the normal distribution of forces 
in the body, the bones tend to follow the lines of stress, and 
especially the pressure-lines. To demonstrate this in a compre- 
hensive way would doubtless be difficult; for we should be dealing 
with a framework of very great complexity, and should have to 
take account of a great variety of conditions*. This framework 
is complicated as we see it in the skeleton, where (as we have said) 
it is only, or chiefly, the struts of the whole fabric which .are 
represented; but to understand the mechanical structure in 
detail, we should have to follow out the still more complex 
arrangement of the tres, as represented by the muscles and 
ligaments, and we should also require much detailed information 
as to the weights of the various parts and as to the other forces 
concerned. Without these latter data we can only treat the 
question in a preliminary and imperfect way. But, to take once 
again a small and simplified part of a big problem, let us think 
of a quadruped (for instance, a horse) in a standing posture, and 
see whether the methods and terminology of the engineer may not 
help us, as they did in regard to the minute structure of the single 
bone. ; 

Standing four-square upon its forelegs and hindlegs, with the 
weight of the body suspended between, the quadruped at once 
suggests to us the analogy of a bridge, carried by its two piers. 
And if it occurs to us, as naturalists, that we never look at a 
standing quadruped without contemplating a bridge, so, con- 
versely, a similar idea has occurred to the engineer: for Professor 
Fidler, in this Treatise on Bridge-Construction, deals with the chief 
descriptive part of his subject under the heading of “The Com- 
parative Anatomy of Bridges.” The designation is most just, for 
in studying the various types of bridges we are studying a series 
of well-planned skeletons}; and (at the cost of a little pedantry) 

* Our problem is analogous to Dr Thomas Young's problem of the best disposi- 
tion of the timbers in a wooden ship (Phil. Trans. 1814, p. 303). He was not long 
of finding that the forces which may act upon the fabric are very numerous and 
very variable, and that the best mode of resisting them, or best structural arrange- 


ment for ultimate strength, becomes an immensely complicated problem. 
+ In like manner, Clerk Maxwell could not help employing the term “skeleton ”’ 


44—2 


692 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


we might go even further, and study (after the fashion of the 
anatomist) the “osteology” and “desmology” of the structure, 
that is to say the bones which are represented by “struts,” and 
the ligaments, etc., which are represented by “ties.” Further- 
more after the methods of the comparative anatomist, we may 
classify the families, genera and species of bridges according to 
their distinctive mechanical features, which correspond to certain 
definite conditions and functions. 

In more ways than one, the quadrupedal bridge is a remarkable 
one; and perhaps its most remarkable peculiarity is that it is a 
jomted and flexible bridge, remaining in equilibrium under 
considerable and sometimes great modifications of its curvature, 
such as we see, for instance, when a cat humps or flattens her 
back. The fact that flexibility is an essential feature in the 
quadrupedal bridge, while it is the-last thing which an engineer 
desires and the first which he seeks to provide against, will impose 
certain important limiting conditions upon the design of the 
skeletal fabric; but to this matter we shall afterwards return. 
Let us begin by considering the quadruped at rest, when he stands 
upright and motionless upon his feet, and when his legs exercise 
no function save only to carry the weight of the whole body. - So 
far as that function is concerned, we might now perhaps compare 
the horse’s legs with the tall and slender piers of some railway 
bridge; but it is obvious that these jointed legs are ill-adapted 
to receive the horizontal thrust of any arch that may be placed 
atop of them. Hence it follows that the curved backbone of the 
horse, which appears to cross like an arch the span between his 
shoulders and his flanks, cannot be regarded as an arch, in the 


? 


in defining the mathematical conception of a “‘frame,” constituted by points and 
their interconnecting lines: in studying the equilibrium of which, we consider its 
different points as mutually acting on each other with forces whose directions are 
those of the lines joining each pair of points. Hence (says Maxwell), “in order to 
exhibit the mechanical action of the frame in the most elementary manner, we may 
draw it as a skeleton, in which the different points are joined by straight lines, 
and we may indicate by numbers attached to these lines the tensions or com- 
pressions in the corresponding pieces of the frame” (Zrans. R. S. H. xxvi, p. 1, 
1870). It follows that the diagram so constructed represents a ‘“‘diagram of 
forees,”’ in this limited sense that it is geometrical as regards the position and 
direction of the forces, but arithmetical as regards their magnitude. It is to just 
such a diagram that the animal’s skeleton tends to approximate. 


xvi] THE COMPARATIVE ANATOMY OF BRIDGES 693 


engineer’s sense of the word. It resembles an arch in form, but 
not in function, for it cannot act as an arch unless it be held back 
at each end (as every arch is held back) by abutments capable of 
resisting the horizontal thrust; and these necessary abutments 
are not present in the structure. But in various ways the 
engineer can modify his superstructure so as to supply the place 
of these external reactions, which in the simple arch are obviously 
indispensable. Thus, for example, we may begin by inserting a 
straight steel tie, dB (Fig. 339), uniting the ends of the curved rib 
AaB; and this tie will supply the place of the external reactions, 
converting the structure into a “tied arch,” such as we may see 
in the roofs of many railway-stations. Or we may go on to fill 
in the space between arch and tie by a “web-system,” converting 
it into what the engineer describes as a “parabolic bowstring 


a 


Fig. 339. 


girder” (Fig. 3396). In either case, the structure becomes an 
independent “detached girder,” supported at each end but not 
otherwise fixed, and consisting essentially of an upper compression- 
member, AaB, and a lower tension-member, 48. But again, in 
the skeleton of the quadruped, the necessary tie, AB, is not to be 
found; and it follows that these comparatively simple types of 
bridge do not correspond to, nor do they help us to understand, 
the type of bridge which nature has designed in the skeleton of 
the quadruped. Nevertheless if we try to look, as an engineer 
would look, at the actual design of the animal skeleton and the 
actual distribution of its load, we find that the one is most admir- 
ably adapted to the other, according to the strict principles of 
engineering construction. The structure is not an arch, nor a 
tied arch, nor a bowstring girder: but it is strictly and beautifully 


694 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


comparable to the main girder of a double-armed cantilever 
bridge. 

Obviously, in our quadrupedal bridge, the superstructure does 
not terminate (as it did in our former diagram) at the two points 
of support, but it extends beyond them at each end, carrying the 
head at one end and the tail at the other, upon a pair of projecting 
arms or “cantilevers” (Fig. 346). 

In a typical cantilever bridge, such as the Forth Bridge 
(Fig. 345), a certain simplification is introduced. For each pier 
carries, in this case, its own double-armed cantilever, linked by 
a short connecting girder to the next, but so joimted to it that no 
weight is transmitted from one cantilever to another. The bridge 
in short is cut into separate sections, practically imdependent of 
one another; at the joints a certain amount of bending is not 
precluded, but shearing strain is evaded; and each pier carries 
only its own load. By this arrangement the engineer finds that 
design and construction are alike simplified and facilitated. In 
the case of the horse, it is obvious that the two piers of the bridge, 
that is to say the fore-legs and the hind-legs, do not bear (as they 
do in the Forth Bridge) separate and independent loads, but the 
whole system forms a continuous structure. In this case, the 
calculation of the loads will be a little more difficult and the 
corresponding design of the structure a little more complicated. 
We shall accordingly simplify our problem very considerably if, 
to begin with, we look upon the quadrupedal skeleton as con- 
stituted of two separate systems, that is to say of two balanced 
cantilevers, one supported on the fore-legs and the other on the 
hind; and we may deal afterwards with the fact that these two 
cantilevers are not independent, but are bound up in one common 
field of force and plan of construction. 

In the horse it is plain that the two cantilever systems into 
which we may thus analyse the quadrupedal bridge are unequal 
in magnitude and importance. The fore-part of the animal is 
much bulkier than its hind quarters, and the fact that the fore-legs 
carry, as they so evidently do, a greater weight than the hind-legs 
has long been known and is easily proved; we have only to walk 
a horse onto a weigh-bridge, weigh first his fore-legs and then his 
hind-legs, to discover that what we may call his front half weighs 


Xv1| THE SKELETON AS A BRIDGE 695 


a good deal more than what is carried on his hind feet, say about 
three-fifths of the whole weight of the animal. 

The great (or anterior) cantilever then, in the horse, is con- 
stituted by the heavy head and still heavier neck on one side of 
the pier which is represented by the fore-legs, and by the dorsal 
vertebrae carrying a large part of the weight of the trunk upon 
the other side; and this weight is so balanced over the fore-legs 
that the cantilever, while “anchored” to the other parts of the 
structure, transmits but little of its weight to the hind-legs, and 
the amount so transmitted will vary with the position of the 
head and with the position of any artificial load*. Under certain 
conditions, as when the head is thrust well forward, it is evident 
that the hind-legs will be actually relieved of a portion of the 
comparatively small load which is their normal share. 

Our problem now is to discover, in a rough and approximate 
way, some of the structural details which the balanced load upon 
the double cantilever will impress upon the fabric. 


Working by the methods of graphic statics, the engineer’s 
task is, in theory, one of great simplicity. He begins by drawing 
in outline the structure which he desires to erect; he calculates 
the stresses and bending-moments necessitated by the dimensions 
and load on the structure; he draws a new diagram representing 
these forces, and he designs and builds his fabric on the lines of this 
statical diagram. He does, in short, precisely what we have seen 
nature doing in the case of the bone. For if we had begun, as 
it were, by blocking out the femur roughly, and considering its 
position and dimensions, its means of support and the load which 
it has to bear, we could have proceeded at once to draw the system 
of stress-lines which must occupy the field of force: and to 
precisely these stress-lines has nature kept in the building of the 
bone, down to the minute arrangement of its trabeculae. 

The essential function of a bridge is to stretch across a certain 
span, and carry a certain definite load; and this being so, the 

* When the jockey crouches over the neck of his ra¢e-horse, and when Tod 
Sloan introduced the “American seat,’’ the object in both cases is to relieve the 
hind-legs of weight, and so leave them free for the work of propulsion. Never- 


theless, we must not exaggerate the share taken by the-hind-limbs in this latter 
duty; cf. Stillman, The Horse in Motion, p. 69, 1882. 


696 ON FORM AND MECHANICAL EFFICIENCY — [cu. 


chief problem in the designing of a bridge is to provide due 
resistance to the “bending-moments” which result from the load. 
These bending-moments will vary from point to point along the 
girder, and taking the simplest case of a uniform load supported 
at both ends, they will be represented by points on a parabola. 
If the girder be of uniform depth, that is to say if its two flanges, 


oy 


Fig. 340. A, Span of proposed bridge. B, Stress diagram, or diagram 
of bending-moments*. 


respectively under tension and compression, be parallel to one 
another, then the stress upon these flanges will vary as the bending- 
moments, and will accordingly be very severe in the middle and 
will dwindle towards the ends. But if we make the depth of the 
girder everywhere proportional to the bending-moments, that is 


Fig. 341. The bridge constructed, as a parabolic girder. 


to say if we copy in the girder the outlines of the bending-moment 
diagram, then our design will automatically meet the circum- 
stances of the case, for the horizontal stress in each flange will 
now be uniform throughout the length of the girder. In short, in 


* This and the following diagrams are borrowed and adapted from Professor 
Fidler’s Bridge Construction. 


Xvi] OF RECIPROCAL DIAGRAMS 697 


Professor Fidler’s words, “Every diagram of moments represents 
the outline of a framed structure which will carry the given load 
with a uniform horizontal stress in the principal members.” 

In the following diagrams (Fig. 342, a, b) (which are taken 
from the original ones of Cul- 
mann), we see at once that the 
loaded beam or bracket (a) has 
a “danger-point” close to its 
fixed base, that is to say at the 
point remotest from its load. 
But in the parabolic bracket 
(b) there is no danger-point at Fig. 342. 
all, for the dimensions: of the 
structure are made to increase pari passu with the bending- 
moments: stress and resistance vary together. Again in Fig. 340, 
we have a simple span (A), with its stress diagram (B); and in 
Fig. 341 we have the corresponding parabolic girder, whose 
stresses are now uniform throughout. In fact we see that, by a 
process of conversion, the stress diagram in each case becomes 
the structural diagram in the other*. Now all this is but the 
modern rendering of one of Galileo’s most famous propositions. 
In the Dialogue which we have already quoted more than oncey, 
Sagredo says “It would be a fine thing if one could discover the 
proper shape to give a solid in order to make it equally resistant 
at every point, in which case a load placed at the middle would 
not produce fracture more easily than if placed at any other 
point.” And Galileo (in the person of Salviati) first puts the 
problem into its more general form; and then shews us how, by 
giving a parabolic outline to our beam, we have its simple and 
comprehensive solution. 

In the case of our cantilever bridge, we shew the primitive girder 


* The method of constructing reciprocal diagrams, in which one should represent 
the outlines of a frame, and the other the system of forces necessary to keep it 
in equilibrium, was first indicated in Culmann’s Graphische Statik; it was greatly 
developed soon afterwards by Macquorn Rankine (Phil. Mag. Feb. 1864, and 
Applied Mechanics, passim), to whom is mainly due the general application of the 
principle to engineering practice. 

7 Dialogues concerning Two New Sciences (1638): Crew and Salvio’s translation, 
p- 140 seg. 


698 ON FORM AND MECHANICAL EFFICIENCY  [cz. 


in Fig. 343, A, with its bending-moment diagram (B); and it is 
evident that, if we turn this diagram upside down, it will still be 
illustrative, just as before, of the bending-moments from point 
to point: for as yet it is merely a diagram, or graph, of relative 
magnitudes. 

To either of these two stress diagrams, direct or inverted, we 
may fit the design of the construction, as in Figs. 343, C and 344. 


Fig. 344 


Now in different animals the amount and distribution of the 
load differs so greatly that we can expect no single diagram, 
_ drawn from the comparative anatomy of bridges, to apply equally 
well to all the cases met with in the comparative anatomy of 
quadrupeds; but nevertheless we have already gained an insight 
into the general principles of “structural design” in the quad- 
rupedal bridge. 

~ In our last diagram the upper member of the cantilever is under 


Xv1| OF RECIPROCAL DIAGRAMS 699 


tension; it is represented in the quadruped by the lgamentum 
nuchae on the one side of the cantilever, and by the supraspinous 
ligaments of the dorsal vertebrae on the other. The compression 
member is similarly represented, on both sides of the cantilever, 
by the vertebral column, or rather by the bodies of the vertebrae ; 
while the web, or “filling,” of the girders, that is to say the upright 
or sloping members which extend from one flange to the other, is 
represented on the one hand by the spines of the vertebrae, and 
on the other hand, by the oblique interspinous ligaments and 
muscles. The high spines over the quadruped’s withers are no 
other than the high struts which rise over the supporting piers 
in the parabolic girder, and correspond to the position of the 
maximal bending-moments. The fact that these tall vertebrae 
of the withers usually slope backwards, sometimes steeply, in 
a quadruped, is easily and obviously explained*. For each 
vertebra tends to act as a “hinged lever,” and its spine, acted 
on by the tensions transmitted by the ligaments on either side, 
takes up its position as the diagonal of the parallelogram of 
forces to which it is exposed. 

It happens that im these comparatively simple types of 
cantilever bridge the whole of the parabolic curvature is trans- 
ferred to one or other of the principal members, either the 
tension-member or the compression-member as the case may be. 
But it is of course equally permissible to have both members 
curved, in opposite directions. This, though not exactly the case 
in the Forth Bridge, is approximately so; for here the main 
compression-member is curved or arched, and the main tension- 
member slopes downwards on either side from its maximal height 
above the piers. In short, the Forth Bridge is a nearer approach 
than either of the other cantilever bridges which we have 

* The form and direction of the vertebral spines have been frequently and 
elaborately described; cf. (e.g.) Gottlieb, H., Die Anticlinie der Wirbelsiule der 
Saugethiere, Morphol. Jahrb. ux1x, pp. 179-220, 1915, and many works quoted 
therein. According to Morita, Ueber die Ursachen der Richtung und Gestalt der 
thoracalen Dornfortsaitze der Saugethierwirbelsiule (7bz cit. p. 201), various changes 
take place in the direction or inclination of these processes in rabbits, after section 
of the interspinous ligaments and muscles. These changes seem to be very much 
what we should expect, on simple mechanical grounds. See also Fischer, O., 


Theoretische Grundlagen fiir eine Mechanik der lebenden Korper, Leipzig, pp. x. 
372, 1906. 


700 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


illustrated to the plan of the quadrupedal skeleton; for the main 
compression-member almost exactly recalls the form of the verte- 
bral column, while the main tension-member, though not so 
closely similar to the supraspinous and nuchal ligaments, corre- 
sponds to the plan of these in a somewhat simplified form. 


Fig. 345, A two-armed cantilever of the Forth Bridge. Thick lines, com- 
pression-members (bones); thin lines, tension-members (ligaments). 


We may now pass without difficulty from the two-armed 
cantilever supported on a single pier, as it is in each separate 
section of the Forth Bridge, or as we have imagined it to be in 
the forequarters of a horse, to the condition which actually exists 
in that quadruped, where a two-armed cantilever has its load 
distributed over two separate piers. This is not precisely what 
an engineer calls a “continuous” girder, for that term is applied 
to a girder which, as a continuous structure, crosses two or more 
spans, while here there is only one. But nevertheless, this girder 


Fig. 346. 


is effectively continuous from the head to the tip of the tail; and 
at each point of support (4 and B) it is subjected to the negative 
bending-moment due to the overhanging load on each of the 
projecting cantilever arms AH and BT. The diagram of bending- 
moments will (according to the ordinary conventions) lie below 


XVI] OF RECIPROCAL DIAGRAMS 701 


the base line (because the moments are negative), and must take 
some such form as that shown in the diagram: for the girder 
must suffer its greatest bending stress not at the centre, but at 
the two points of support 4 and B, where the moments are 
measured by the vertical ordinates. It is plain that this figure 
only differs from a representation of two independent two-armed 
cantilevers in the fact that there is no point midway in the span 
where the bending-moment vanishes, but only a region between 
the two piers in which its magnitude tends to diminish. 

The diagram effects a graphic summation of the positive and 
negative moments, but its form may assume various modifications 
according to the method of graphic summation which we may 
choose to adopt; and it is obvious also that the form of the 
diagram may assume many modifications of detail according to 
the actual distribution of the load. In all cases the essential 
points to be observed are these: firstly that the girder which is 


ae | ae 


B A 
Tail Head 


Fig. 347. Stress-diagram of horse’s backbone. 


to resist the bending-moments induced by the load must possess 
its two principal members—an upper tension-member or tie, 
represented by ligament, and a lower compression-member 
represented by bone: these members being united by a web 
represented by the vertebral spines with their interspinous liga- 
ments, and being placed one above the other in the order named 
because the moments are negative; secondly we observe that the 
depth of the web, or distance apart of the principal members,— 
that is to say the height of the vertebral spines,—must be pro- 
portional to the bending-moment at each point along the length 
of the girder. 

In the case of an animal carrying two-thirds of his weight 
upon his fore-legs and only one-third upon his hind-legs, the 
bending-moment diagram will be unsymmetrical, after the fashion 
of Fig. 347, the vertical ordinate at A being thrice the height of 
that at B. 


702 ON FORM AND MECHANICAL EFFICIENCY  [ca. 


On the other hand the Dinosaur, with his light head and 
enormous tail would give us a moment-diagram with the opposite 
kind of asymmetry, the greatest bending stress bemg now found 
oyer the haunches, at B (Fig. 348). A glance at the skeleton of 
Diplodocus Carnegv will shew us the high vertebral spines over 
the loins, in precise correspondence with the requirements of this 
diagram: just as in the horse, under the opposite conditions of 
load, the highest vertebral spines are those of the withers, that 
is to say those of the posterior cervical and anterior dorsal 
vertebrae. 

We have now not only dealt with the general resemblance, 
both in structure and in function, of the quadrupedal backbone 
with its associated ligaments to a double-armed cantilever girder, 
but we have begun to see how the characters of the vertebral 
system must differ in different quadrupeds, according to the 


imi 


B A 
Tail Head 


Fig. 348. Stress-diagram of backbone of Dinosaur. 


conditions imposed by the varying distribution of the load: and 
in particular how the height of the vertebral spines which con- 
stitute the web will be in a ‘definite relation, as regards magnitude 
and position, to the bending-moments induced thereby. We 
should require much detailed information as to the actual weights 
of the several parts of the body before we could follow out 
quantitatively the mechanical efficiency of each type of skeleton; 
but in an approximate way what we have already learnt will 
enable us to trace many interesting correspondences between 
structure and function in this particular part of comparative 
anatomy. We must, however, be careful to note that the great 
cantilever system is not of necessity constituted by the vertical 
column and its ligaments alone. but that the pelvis, firmly united 
as it is to the sacral vertebrae, and stretching backwards far 
beyond the acetabulum, becomes an intrinsic part of the system ; 
and helping (as it does) to carry the load of the abdominal viscera, 


/ 


XVI] IN THE SKELETON OF QUADRUPEDS 703 


constitutes a great portion of the posterior cantilever arm, or 
even its chief portion in cases where the size and weight of the 
tail are insignificant, as is the case in the majority of terrestrial 
mammals. 

We mav also note here, that just as a bridge is often a 
“combined” or composite structure, exhibiting a combination of 
principles in its construction, so in the quadruped we have, as 
it were, another girder supported by the same piers to carry the 
viscera; and consisting of an inverted parabolic girder, whose 
compression-member is again constituted by the backbone, its 
tension-member by the line of the sternum and the abdominal 
muscles, while the ribs and intercostal muscles play the part of 
the web or filling. 

A very few instances must suffice to illustrate the chief 
variations in the load, and therefore in the bending-moment 
diagram, and therefore also in the plan of construction, of various 
quadrupeds. But let us begin by setting forth, in a few cases, 
the actual weights which are borne by the fore-limbs and the 
hind-limbs, in our quadrupedal bridge*. 


On On Hy Gral Se Oa 

Gross weight. Fore-feet. Hind-feet Fore- Hind- 

ton cwts. ewts, ewts. feet. feet. 

Camel (Bactrian) — 14-25 9-25 4-5 67:3 32-7 

Llama 7275 S75 875 66-7 33:3 

Elephant (Indian) 1 15:75 20-5 14-75 58-2 41-8 

Horse — 8:25 4:75 35 57-6 42-4 
Horse (large Clydes- 

dale) — 15:5 8-5 7:0 54:8 45-2 


7 


It will be observed that in all these animals the load upon the 
fore-feet preponderates considerably over that upon the hind, the 
preponderance being rather greater in the elephant than in the 
horse, and markedly greater in the camel and the llama than in 
the other two. But while these weights are helpful and sug- 
gestive, it is obvious that they do not go nearly far enough to 
give us a full insight into the constructional diagram to which 
the animals are conformed. For such a purpose we should 

* T owe the first four of these determinations to the kindness of Dr Chalmers 


Mitchell, who had them made for me at the Zoological Society’s Gardens; while 
the great Clydesdale carthorse was weighed for me by a friend in Dundee. 


704. ON FORM AND MECHANICAL EFFICIENCY  [cu. 


require to weigh the total load, not in two portions, but i many; 
and we should also have to take close account of the general form 
of the animal, of the relation between that form and the distribu- 
tion of the load, and of the actual directions of each bone and 
ligament by which the forces of compression and tension were 
transmitted. All this hes beyond us for the present; but never- 
theless we may consider, very briefly, the principal cases involved 
in our enquiry, of which the above animals form only a partial 
and preliminary illustration. 

(1) Wherever we have a heavily loaded anterior cantilever 
arm, that is to say whenever the head and neck represent a 
considerable fraction of the whole weight of the body, we tend 
to have large bending-moments over the fore-legs, and corre- 
spondingly high spines over the vertebrae of the withers. This 


Tail Head 
a 


A 


Fig. 349. Stress-diagram of Titanotherium. 


is the case in the great majority of four-footed, terrestrial animals, 
the chief exceptions being found in animals with comparatively 
small heads but large and heavy tails, such as the anteaters or 
the Dinosaurian reptiles, and also (very naturally) in animals 
such as the crocodile, where the “bridge” can scarcely be said 
to be developed, for the long heavy body sags down to rest upon 
the ground. The case is sufficiently exemplified by the horse, 
and still more notably by the stag, the ox, or the pig. It is 
illustrated in the accompanying diagram of the conditions in the 
great extinct Titanotherium. 

(2) In the elephant and the camel we have similar conditions, 
but slightly modified. In both cases, and especially in the latter, 
the weight on the fore-quarters is relatively large; and in both 
cases the bending-moments are all the larger, by reason of the 
length and forward extension of the camel’s neck, and the forward 


XVI] IN THE SKELETON OF QUADRUPEDS 705 


position of the heavy tusks of the elephant. In both cases the _ 
dorsal spines are large, but they do not strike us as exceptionally 
so; but in both cases, and especially in the elephant, they slope 
backwards in a marked degree. Lach spine, as already explained, 
must in all cases assume the position of the diagonal in the 
parallelogram of forces defined by the tensions acting on it at 
its extremity; for it constitutes a “hinged lever,” by which the 
bending-moments on either side are automatically balanced; and 
it is plain that the more the spine slopes backwards the more it 
indicates a relatively large strain thrown upon the great ligament 
of the neck, and a relief of strain upon the more directly acting, 
but weaker, ligaments of the back and loins. In both cases, the 
bending-moments would seem to be more evenly distributed over 
the region of the back than, for instance, in the stag, with its 
light hind-quarters and heavy load of antlers: and in both cases _ 
the high “girder” is considerably prolonged, by an extension of 
the tall spines backwards in the direction of the lois. When 
we come to such a case as the mammoth, with its immensely 
heavy and immensely elongated tusks, we perceive at once that 
the bending-moments over the fore-legs are now very severe; 
and we see also that the dorsal spines in this region are much 
more conspicuously elevated than in the ordinary elephant. 

(3) In the case of the giraffe we have, without doubt, a very 
heavy load upon the fore-legs, though no weighings are at hand 
to define the ratio; but as far as possible this disproportionate 
load would seem to be relieved, by help of a downward as well 
as backward thrust, through the sloping back, to the unusually 
low hind-quarters. The dorsal spines of the vertebrae are very 
high and strong, and the whole girder-system very perfectly 
- formed. The elevated, rather than protruding position of the 
head lessens the anterior bending-moment as far as possible; but. 
it leads to a strong compressional stress transmitted almost 
directly downwards through the neck: in correlation with which 
we observe that the bodies of the cervical vertebrae are excep- 
tionally large and strong and steadily increase in size and strength 
from the head downwards. 

(4) In the kangaroo, the fore-limbs are entirely relieved of 
their load, and accordingly the tall spines over the withers, which 


TGs 45 


706 ON FORM AND MECHANICAL EFFICIENCY  [cu. 


were so conspicuous in all heavy-headed quadrupeds, have now 
completely vanished. The creature has become bipedal, and body 
and tail form the extremities of a single balanced cantilever, 
whose maximal bending-moments are marked by strong, high 
lumbar and sacral vertebrae, and by iliac bones of peculiar — 
and exceptional strength. 

Precisely the same condition is illustrated in the Jguanodon, 
and better still by reason of the great bulk of the creature, and of 
the heavy load which falls to be supported by the great cantilever 
and by the hind-legs which form its piers. The long and heavy 
body and neck require a balance-weight (as in the kangaroo) in 
the form of a long heavy tail. And the double-armed cantilever, 
so constituted, shews a beautiful parabolic curvature in the graded 
heights of the whole series of vertebral spines, which rise to a 
maximum over the haunches and die away slowly towards the 
neck and the tip of the tail. 

(5) In the case of some of the great pees fossil reptiles, 
such as Diplodocus, it has always been a more or less disputed 
question whether or not they assumed, like Iguanodon, an erect, 
bipedal attitude. In all of these we see an elongated pelvis, and, 
in still more marked degree, we see elevated spinous processes of 
the vertebrae over the hind-limbs; in all of them we have a long 
heavy tail, and in most of them we have a marked reduction in 
size and weight both of the fore-limb and of the head itself. The 
great size of these animals is not of itself a proof against the erect 
attitude; because it might well have been accompanied by an 
aquatic or partially submerged habitat, and the crushing stress of 
the creature’s huge bulk proportionately relieved. But we must 
consider each such case in the whole light of its own evidence; ~ 
and it is easy to see that, just as the quadrupedal mammal may 
carry the greater part but not all of its weight upon its fore-limbs, 
so a heavy-tailed reptile may carry the greater part upon its hind- 
limbs, without this process going so far as to relieve its fore-limbs 
of all weight whatsoever. This would seem to be the case in such 
a form as Diplodocus, and also in Stegosaurus, whose restoration 
by Marsh is doubtless substantially correct*. The fore-limbs, 


* This pose of Diplodocus, and of other Sauropodous reptiles, has been much 
discussed. Cf. (int. al.) Abel, O., Abh. k. k. zool. bot. Ges. Wien, v. 1909-10 (60 pp.) ; 


Xvi] IN THE SKELETON OF QUADRUPEDS 707 


though comparatively small, are obviously fashioned for support, 
but the weight which they have to carry is far less than that 
which the hind-limbs bear. The head is small and the neck 
short, while on the other hand the hind-quarters and the tail are 
big and massive. The backbone bends into a great, double-armed 
cantilever, culminating over the pelvis and the hind-limbs, and 
here furnished with its highest and strongest spines to separate 
the tension-member from the compression-member of the girder. 
The fore-legs form a secondary supporting pier to this great 
cantilever, the greater part of whose weight is poised upon the 
hind-limbs alone. 


Fig. 350. Diagram of Stegosaurus. 


(6) In a bird, such as an ostrich or a common fowl, the 
bipedal habit necessitates the balancing of the load upon a single 
double-armed cantilever-girder, just as in the Iguanodon and the 
kangaroo, but the construction is effected in a somewhat different 
way. The great heavy tail has entirely disappeared; but, though 
from the skeleton alone it would seem that nearly all the bulk of 
the animal lay in front of the hind-limbs, yet in the living bird 
we can easily perceive that the great weight of the abdominal 
organs lies suspended behind the socket for the thigh-bone, and 
so hangs from the posterior lever-arm of the cantilever, balancing 
the head and neck and thorax whose combined weight hangs from 


Tornier, SB. Ges. Naturf. Fr. Berlin, pp. 193-209, 1909; Hay, O. P., Amer. Nat. 
Oct. 1908; Tr. Wash. Acad. Sci. x11, pp. 1-25, 1910; Holland, Amer. Nai. May, 
1910, pp. 259-283; Matthew, ibid. pp. 547-560; Gilmore, C. W. (Restoration of 
Stegosaurus), Pr. U.S. Nat. Museum, 1915. 


45—2 


708 ON FORM AND MECHANICAL EFFICIENCY [CH. 


the anteriorarm. The great cantilever girder appears, accordingly, 
balanced over the hind-legs. It is now constituted in part by 
the posterior dorsal or lumbar vertebrae, all traces of special 
elevation having disappeared from the anterior dorsals; but the 
greater part of the girder is made up of the great iliac bones, 
placed side by side, and gripping firmly the sacral vertebrae, often 
almost to the extinction of these latter. In the form of these 
iliac bones, the arched curvature of their upper border, in their 
elongation fore-and-aft to overhang both ways their supporting 
pier, and in the coincidence of their greatest height with the 
median line of support over the centre of gravity, we recognise 
all the characteristic properties of the typical balanced canti- 
lever=, 

(7) We find a highly important corollary in the case of 
aquatic animals. For here the effect of gravity is neutralised ; 
we have neither piers nor cantilevers; and we find accordingly 
in all aquatic mammals of whatsoever group—whales, seals or 
sea-cows—that the high arched vertebral spines over the withers, 
or corresponding structures over the hind-limbs, have both 
entirely disappeared. 

Just as the cantilever girder tended to become obsolete in the 
aquatic mammal so does it tend to weaken and disappear in the 
aquatic bird. There is a very marked contrast between the high- 
arched strongly-built pelvis in the ostrich or the hen, and the 
long, thin, comparatively straight and weakly bone which repre- 
sents it in a diver, a grebe or a penguin. 

But in the aquatic mammal, such as a whale or a dolphin (and 
not less so in the aquatic bird), steffness must be ensured in order 
to enable the muscles to act against the resistance of the water 
in the act of swimming; and accordingly nature must provide 
against bending-moments irrespective of gravity. In the dolphin, 
at any rate as regards its tail end, the conditions will be not very 
different from those of a column or beam with fixed ends, in 
which, under deflexion, there will be two points of contrary 
flexure, as at C, D, in Fig. 351. 


* The form of the cantilever is much less typical in the small flying birds, 
where the strength of the pelvic region is insured in another way, with which we 
need not here stop to deal. 


Xvi] IN AQUATIC ANIMALS 709 


Here, between C and D we have a varying bending-moment, 
represented by a continuous curve with its maximal elevation 
midway between the points of inflexion. And correspondingly, 


in our dolphin, we have a con- E 

tinuous series of high dorsal ,yp 6 De B 

spines, rising to a maximum F 

about the middle of the animal’s © H 
Fig. 351. 


body, and falling to nil at some 
distance from the end of the tail. It is their business (as 
usual) to keep the tension-member, represented by the strong 
supraspinous ligaments, wide apart from the compression-member, 
which is as usual represented by the backbone itself. But in 
our diagram we see that on the further side of C and D we 
have a negative curve of bending-moments, or bending-moments 
in a contrary direction. Without inquiring how these stresses 
are precisely met towards the dolphin’s head (where the coalesced 
cervical vertebrae suggest themselves as a partial explanation), 
we see at once that towards the tail they are met by the strong 
series of chevron-bones, which in the caudal region, where tall 
dorsal spines are no longer needed, take their place below the 
vertebrae, in precise correspondence with the bending-moment 
diagram. In many cases other than these aquatic ones, when 
we have to deal with animals with long and heavy tails (like the 
Iguanodon and the kangaroo of which we have already spoken), 
we are apt to meet with similar, though usually shorter chevron- 
bones; and in all these cases we may see without difficulty that 
a negative bending-moment is there to be resisted. 

In the dolphin we may find a good illustration of the fact 
that not only is it necessary to provide for rigidity in the vertical 
direction, but also in the horizontal, where a tendency to bending 
must be resisted on either side. This function is effected in part 
by the ribs with their associated muscles, but they extend but a 
little way and their efficacy for this purpose can be but small. 
We have, however, behind the region of the ribs and on either side 
of the backbone a strong series of elongated and flattened trans- 
verse processes, forming a web for the support of a tension-member 
in the usual form of ligament, and so playing a part precisely 
analogous to that performed by the dorsal spines in the same 


710 ON FORM AND MECHANICAL EFFICIENCY  [cn. 


animal. In an ordinary fish, such as a cod or a haddock, we see 
precisely the same thing: the backbone is stiffened by the indis- 
pensable help of its three series of ligament-connected processes, 
the dorsal and the two transverse series. And here we see (as 
we see partly also among the whales), that these three series of — 
processes, or struts, tend to be arranged well-nigh at equal angles, 
of 120°, with one another, giving the greatest and most uniform 
strength of which such a system is capable. On the other hand, 
‘in a flat fish, such as a plaice, where from the natural mode of 
progression it is necessary that the backbone should be flexible 
in one direction while stiffened in another, we find the whole 
outline of the fish comparable to that of a double bowstring 
girder, the compression-member being (as usual) the backbone, 
the tension-member on either side being constituted by the inter-_ 
spinous ligaments and muscles, while the web or filling is very 
beautifully represented by the long and evenly graded spines, 
which spring symmetrically from opposite sides of each individual 
- vertebra. 


The main result at which we have now arrived, in regard to 
the construction of the vertebral column and its associated parts 
is that we may look upon it as a certain type of gurder, whose depth, 
as we cannot help seeing, is everywhere very nearly proportional 
to the height of the corresponding ordinate in the diagram of 
moments: just as it is in the girder of a cantilever bridge as 
designed by a modern engineer. In short, after the nineteenth 
or twentieth century engineer has done his best in framing the 
design of a big cantilever, he may find that some of his best ideas” 
had, so to speak, been anticipated ages ago in the fabric of the 
great saurians and the larger mammals. 

But it is possible that the modern engineer might be disposed 
to criticise the skeleton girder at two or three points; and in 
particular he might think the girder, as we see it for instance i 
Diplodocus or Stegosaurus, not deep enough for carrying the 
animal’s enormous weight of some twenty tons. If we adopt aq 
much greater depth (or ratio of depth to length) as in the modern | 
cantilever, we shall greatly increase the strength of the structure; 
but at the same time we should greatly increase its rigidity, and 


XVI] ON STRENGTH AND FLEXIBILITY 711 


this is precisely what, in the circumstances of the case, it would 
seem that nature is bound to avoid. We need not suppose that 
the great saurian was by any means active and limber; but a 
certain amount of activity and flexibility he was bound to have, 
and in a thousand ways he would find the need of a backbone 
that should be flexible as well as strong. Now this opens up a 
new aspect of the matter and is the beginning of a long, long story, 
for in every direction this double requirement of strength and 
flexibility imposes new conditions upon our design. To represent 
all the correlated quantities we should have to construct not only 
a diagram of moments but also a diagram of elastic deflexion and 
its so-called “curvature”; and the engineer would want to know 
something more about the material of the ligamentous tension- 
member—its modulus of elasticity in direct tension, its elastic 
limit, and its safe working stress. 

In various ways our structural problem is beset by “limiting 
conditions.” Not only must rigidity be associated with flexibility, 
but also stability must be ensured in various positions and 
attitudes; and the primary function of support or weight-carrying 
must be combined with the provision of points @appui for the 
muscles concerned in locomotion. We cannot hope to arrive at 
a numerical or quantitative solution of this complicate problem, 
but we have found it possible to trace it out in part towards a 
qualitative solution. And speaking broadly we may certainly 
say that in each case the problem has been solved by nature 
herself, very much as she solves the difficult problems of minimal 
areas in a system of soap-bubbles; so that each animal is fitted 
with a backbone adapted to his own individual needs, or (in 
other words) corresponding exactly to the mean resultant of the 
stresses to which as a mechanical system it is exposed. 


Throughout this short discussion of the principles of con- 
struction, limited to one part of the skeleton, we see the same 
general principles at work which we recognise in the plan and 
construction of an individual bone. That is to say, we see a 
tendency for material to be laid down just in the lines of stress, 
and so as to evade thereby the distortions and disruptions due to 
shear. In these phenomena there lies a definite law of growth, 


712 ON FORM AND MECHANICAL EFFICIENCY — [ca. 


whatever its ultimate expression or explanation may come to be. 
Let us not press either argument or hypothesis too far: but be 
content to see that skeletal form, as brought about by growth, 
is to a very large extent determined by mechanical considerations, 
and tends to manifest itself as a diagram, or reflected image, of 
mechanical stress. If we fail, owing to the immense complexity 
of the case, to unravel all the mathematical principles involved 
in the construction of the skeleton, we yet gain something, and 
not a little, by applying this method to the familiar objects of our 
anatomical study: obvia conspicimus, nubem pellente mathesi*. 

Before we leave this subject of mechanical adaptation, let us 
dwell once more for a moment upon the considerations which 
arise from our conception of a field of force, or field of stress, in 
which tension and compression (for instance) are inevitably 
combined, and are met by the materials naturally fitted to resist 
them. It has been remarked over and over again how harmoni- 
ously the whole organism hangs together, and how throughout 
its fabric one part is related and fitted to another im strictly 
functional correlation. But this conception, though never denied, 
is sometimes apt to be forgotten in the course of that process of 
more and more minute analysis by which, for simplicity’s sake, 
we seek to unravel the intricacies of a complex organism. 

We tend, as we analyse a thing into its parts or into its 
properties, to magnify these, to exaggerate their apparent 
independence, and to hide from ourselves (at least for a time) the 
essential integrity and individuality of the composite whole. We 
divide the body into its organs, the skeleton into its bones, as 
in very much the same fashion we make a subjective analysis of 
the mind, according to the teachings of psychology, into component 
factors: but we know very well that judgment and knowledge, 
courage or gentleness, love or fear, have no separate existence, 
but are somehow mere manifestations, or imaginary co-efficients, 
of a most complex integral. And likewise, as biologists, we may 
go so far as to say that even the bones themselves are only in a 
limited and even a deceptive sense, separate and individual 
things. -The skeleton begins as a continuum, and a continuum it 
remains all life long. The things that link bone with bone, 


* The motto was Macquorn Rankine’s. 


Ee 


XVI] ON THE SKELETON AS A WHOLE 713 


cartilage, ligaments, membranes, are fashioned out of the same 
primordial tissue, and come into being pari passu, with the bones 
themselves. The entire fabric has its soft parts and its hard, its 
rigid and its flexible parts; but until we disrupt and dismember 
its bony, gristly and fibrous parts, one from another, it exists 
simply as a “skeleton,” as one integral and individual whole. 
A bridge was once upon a time a loose heap of pillars and rods 
and rivets of steel. But the identity of these is lost, just as if 
they were fused into a solid mass, when once the bridge is built; 
their separate functions are only to be recognised and analysed 
in so far as we can analyse the stresses, the tensions and the 
pressures, which affect this part of the structure or that; and 
these forces are not themselves separate entities, but are the 
resultants of an analysis of the whole field of force. Moreover 
when the bridge is broken it is no: longer a bridge, and all its 
strength is gone. So is it precisely with the skeleton. In it is 
reflected a field of force: and keeping pace, as it were, in action 
and interaction with this field of force, the whole skeleton and 
every part thereof, down to the minute intrinsic structure of the 
bones themselves, is related in form and in position to the lines 
of force, to the resistances it has to encounter; for by one of 
the mysteries of biology, resistance begets resistance, and where 
pressure falls there growth springs up in strength to meet it. 
And, pursuing the same train of thought, we see that all this is 
true not of the skeleton alone but of the whole fabric of the body. 
Muscle and. bone, for instance, are inseparably associated and 
connected; they are moulded one with another; they come into 
being together, and act and react together*. We may study 
them apart, but it is as a concession to our weakness and to the 
narrow outlook of our minds. We see, dimly perhaps, but yet 
with all the assurance of conviction, that between muscle and 
bone there can be no change in the one but it is correlated with 
changes in the other; that through and through they are linked 
in indissoluble association; that they are only separate entities 


* John Hunter was seldom wrong; but I cannot believe that he was right when 
he said (Scientific Works, ed. Owen, 1, p. 371), ‘‘The bones, in a mechanical view, 
appear to be the first that are to be considered. We can study their shape, 
connexions, number, uses, etc., without considering any other part of the body.” 


714. ON FORM AND MECHANICAL EFFICIENCY  [cu. 


in this limited and subordinate sense, that they are parts of a 
whole which, when it loses its composite integrity, ceases to 
exist. 

The biologist, as well as the philosopher, learns to recognise 
that the whole is not merely the sum of its parts. It is this, and 
much more than this. For it is not a bundle of parts but an 
organisation of parts, of parts in their mutual arrangement, 
fitting one with another, in what Aristotle calls “a single and 
indivisible principle of unity”; and this is no merely metaphysical 
conception, but is in biology the fundamental truth which lies at 
the basis of Geoffroy’s (or Goethe’s) law of “compensation,” or 
“balancement of growth.” 

Nevertheless Darwin found no difficulty in believing that 
‘natural selection will tend in the long run to reduce any part 
of the organisation, as soon as, through changed habits, it becomes 
superfluous: without by any means causing some other part to 
be largely developed in a corresponding degree. And conversely, 
that natural selection may perfectly well succeed in largely deve- 
loping an organ without requiring as a necessary compensation 
the reduction of some adjoining part*.” This view has been 
developed into a doctrine of the “independence of single char- 
acters” (not to be confused with the germinal “unit characters” 
of Mendelism), especially by the palaeontologists. Thus Osborn 
asserts a “principle of hereditary correlation,” combined with a 
“principle of hereditary separability whereby the body is a colony, 
a mosaic, of single individual and separable characters}.” 
I cannot think that there is more than a small element of truth 
in this doctrine. As Kant said, “die Ursache der Art der Existenz 
bei jedem Theile eines lebenden Korpers ist om Ganzen enthalten.” 
And, according to the trend or aspect of our thought, we may 
look upon the co-ordinated parts, now as related and fitted to the 


‘ 


end or function of the whole, and now as related to or resulting 


from the physical causes inherent in the entire system of forces 
to which the whole has been exposed, and under whose influence 
it has come into beingt. 


* Origin of Species, 6th ed. p. 118. 
+ Amer. Naturalist, April, 1915, p. 198, ete. Cf. infra, p. 727. 
{ Driesch sees in “Entelechy” that something which differentiates the whole 


Xvi] THE PROBLEM OF PHYLOGENY 715 


It would seem to me that the mechanical principles and 
phenomena which we have dealt with in this chapter are of no small 
impottance to the morphologist, all the more when he is inclined 
to direct his study of the skeleton exclusively to the problem of 
phylogeny; and especially when, according to the methods of 
modern comparative morphology, he is apt to take the skeleton 
to pieces, and to draw from the comparison of a series of scapulae, 
humeri, or individual vertebrae, conclusions as to the descent 
and relationship of the animals to which they belong. 

It would, I dare say, be a gross exaggeration to see in every 
bone nothing more than a resultant of immediate and direct 
physical or mechanical conditions; for to do so would be te deny 
the existence, in this connection, of a principle of heredity. And 
though I have tried throughout this book to lay emphasis on the 
direct action of causes other than heredity, in short to circum- 
scribe the employment of the latter as a working hypothesis in 
morphology, there can still be no question whatsoever but that 
heredity is a vastly important as well as a mysterious thing; it 
is one of the great factors in biology, however we may attempt to 
figure to ourselves, or howsoever we may fail even to imagine, 
its underlying physical explanation. But I maintain that it is 
no less an exaggeration if we tend to neglect these direct physical 
and mechanical modes of causation altogether, and to see in the 
characters of a bone merely the results of variation and of heredity, 
and to trust, in consequence, to those characters as a sure and 
certain and unquestioned guide to affinity and phylogeny. 
Comparative anatomy has its physiological side, which filled 
men’s minds in John Hunter’s day, and in Owen’s day; it has its 


from the sum of its parts in the case of the organism: ‘The organism, we know, 
is a system the single constituents of which are inorganic in themselves; only the 
whole constituted by them in their typical order or arrangement owes its specificity 
to ‘Entelechy’” (Gifford Lectures, p. 229, 19U8): and I think it could be shewn 
that many other philosophers have said precisely the same thing. So far as the 
argument goes, I fail to see how this Entelechy is shewn to be peculiarly or 
specifically related to the living organism. The conception that the whole is 
always something very different from its parts is a very ancient doctrine. The 
reader will perhaps remember how, in another vein, the theme is treated by Martinus 
Scriblerus: ‘‘In every Jack there is a meat-roasting Quality, which neither resides 
in the fly, nor in the weight, nor in any particular wheel of the Jack, but is the 
result of the whole composition; etc., etc.” 


716 ON FORM AND MECHANICAL EFFICIENCY  [ca. 


classificatory and phylogenetic aspect, which has all but filled 
men’s minds during the last couple of generations; and we can 
lose sight of neither aspect without risk of error and misconcéption. 

It is certain that the question of phylogeny, always difficult, 
becomes especially so in cases where a great change of physical 
or mechanical conditions has come about, and where accordingly 
the physical and physiological factors in connection with change 
of form are bound to be large. To discuss these questions at 
length would be to enter on a discussion of Lamarck’s philosophy 
of biology, and of many other things besides. But let us take 
one single illustration. 

The affinities of the whales constitute, as will be readily 
admitted, a very hard problem in phylogenetic classification. 
We know now that the extinct Zeuglodons are related to the 
old Creodont carnivores, and thereby (though distantly) to the 
seals; and it is supposed, but it is by no means so certain, that 
in turn they are to be considered as representing, or as allied to, 
the ancestors of the modern toothed whales*. The proof of any 
such a contention becomes, to my mind, extraordinarily difficult 
and complicated; and the arguments commonly used in such cases 
may be said (in Bacon’s phrase) to allure, rather than to extort 
assent. Though the Zeuglodonts were aquatic animals, we do not 
know, and we have no right to suppose or to assume, that they 
swam after the fashion of a whale (any more than the seal does), 
that they dived like a whale, and leaped like a whale. But the fact 
that the whale does these things, and the way in which he does 
them, is reflected in many parts of his skeleton—perhaps more 
or less in all: so much so that the lines of stress which these 
actions impose are the very plan and working-diagram of great 
part of his structure. That the Zeuglodon has a scapula like that 
of a whale is to my mind no necessary argument that he is akin 
by blood-relationship to a whale: that his dorsal vertebrae are 
very different from a whale’s is no conclusive argument that 

* “There can be no doubt that Fraas is correct in regarding this type (Procetus) 
as an annectant form between the Zeuglodonts and the Creodonta, but, although 
the origin of the Zeuglodonts is thus made clear, it still seems to be by no means 
so certain as that author believes, that they may not themselves be the ancestral 


forms of the Odontoceti”; Andrews, Tertiary Vertebrata of the Fayum, 1906, 
p- 235. 


XVI] THE PROBLEM OF PHYLOGENY raw 


such blood-relationship is lacking. The former fact goes a long 
way to prove that he used his flippers very much as a whale does; 
the latter goes still farther to prove that his general movements 
and equilibrium in the water were totally different. The whale 
may be descended from the Carnivora, or might for that matter, 
as an older school of naturalists believed, be descended from the 
Ungulates; but whether or no, we need not expect to find in him 
the scapula, the pelvis or the vertebral column of the lion or of 
the cow, for it would be physically impossible that he could live 
the life he does with any one of them. In short, when we hope to 
find the missing links between a whale and his terrestrial ancestors, 
it must be not by means of conclusions drawn from a scapula, an 
axis, or even from a tooth, but by the discovery of forms so inter- 
mediate in their general structure as to indicate an organisation 
’ and, zpso facto, a mode of life, intermediate between the terrestrial 
and the Cetacean form. There is no valid syllogism to the effect 
that A has a flat curved scapula like a seal’s, and B has a flat, 
curved scapula like a seal’s: and therefore A and B are related 
to the seals and to each other; it is merely a flagrant case of an 
“undistributed middle.” But there is validity in an argument 
that B shews in its general structure, extending over this bone 
and that bone, resemblances both to A and to the seals: and that 
therefore he may be presumed to be related to both, in his 
hereditary habits of life and in actual kinship by blood. It is 
cognate to this argument that (as every palaeontologist knows) 
we find clues to affinity more easily, that is to say with less 
confusion and perplexity, in certain structures than in others. 
The deep-seated rhythms of growth which, as I venture to 
think, are the chief basis of morphological heredity, bring about 
similarities of form, which endure in the absence of conflicting 
forces; but a new system of forces, introduced by altered environ- 
ment and habits, impinging on those particular parts of the fabric 
which le within this particular field of force, will assuredly not 
be long of manifesting itself in notable and inevitable modifications 
of form. And if this be really so, it will further imply that 
modifications of form will tend to manifest themselves, not so 
much in small and isolated phenomena, in this part of the fabric 
or in that, in a scapula for instance or a humerus: but rather in 


718 ON FORM AND MECHANICAL EFFICIENCY [cn. xvi 


some slow, general, and more or less uniform or graded modification, 
spread over a number of correlated parts, and at times extending 
over the whole, or over great portions, of the body. Whether 
any such general tendency to widespread and correlated trans- 
formation exists, we shall attempt to discuss in the following 
chapter. 


CHAPTER XVII 


ON THE THEORY OF TRANSFORMATIONS, OR THE 
COMPARISON OF RELATED FORMS * 


In the foregoing chapters of this book we have attempted to 
study the inter-relations of growth and form, and the part which 
certain of the physical forces play in this complex interaction ; 
and, as part of the same enquiry, we have tried in comparatively — 
simple cases to use mathematical methods and mathematical 
terminology in order to describe and define the forms of organisms. 
We have learned in so doing that our own study of organic form, 
which we call by Goethe’s name of Morphology, is but a portion 
of that wider Science of Form which deals with the forms assumed 
by matter under all aspects and conditions, and, in a still wider 
sense, with forms which are theoretically imaginable. 

The study of form may be descriptive merely, or it may 
become analytical. We begin by describing the shape of an object 
in the simple words of common speech: we end by defining it 
in the precise language of mathematics; and the one method 
tends to follow the other in strict scientific order and historical 
continuity. Thus, for instance, the form of the earth, of a raindrop 
or a rainbow, the shape of the hanging chain, or the path of a stone 
thrown up into the air, may all be described, however inadequately, 
in common words; but when we have learned to comprehend 
and to define the sphere, the catenary, or the parabola, we have 
made a wonderful and perhaps a manifold advance. The mathe- 
matical definition of a “form” has a quality of precision which 
was quite lacking in our earlier stage of mere description; it is 
expressed in few words, or in still briefer symbols, and these 


* Reprinted, with some changes and additions, from a paper in the Trans. * 
Roy. Soc. Edin. L, pp. 857-95, 1915. 


720 THE THEORY OF TRANSFORMATIONS [CH. 


words or symbols are so pregnant with meaning that thought 
itself is economised; we are brought by means of it in touch with 
Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old 
perhaps as the wisdom of the Egyptians), that “the Book of 
Nature is written in characters of Geometry.” 

Next, we soon reach through mathematical analysis to mathe- 
matical synthesis; we discover homologies or identities which 
were not obvious before, and which our descriptions obscured 
rather than revealed: as for instance, when we learn that, how- 
ever we hold our chain, or however we fire our bullet, the contour 
of the one or the path of the other is always mathematically 
homologous. Lastly, and this is the greatest gain of all, we pass 
quickly and easily from the mathematical conception of form in 
its statical aspect to form in its dynamical relations: we pass from 
the conception of form to an understanding of the forces which. 
gave rise to it; and in the representation of form and in the 
comparison of kindred forms, we see in the one case a diagram 
of forces in equilibrium, and in the other case we discern the 
magnitude and the direction of the forces which have sufficed to 
convert the one form into the other. Here, since a change of 
material form is only effected by the movement of matter, we have 
once again the support of the schoolman’s and the philosopher’s 
axiom, Ignorato motu, ignoratur Natura.” 


In the morphology of living things the use of mathematical 
methods and symbols has made slow progress; and there are 
various reasons for this failure to employ a method whose 
advantages are so obvious in the investigation of other physical 
forms. To begin with, there would seem to be a psychological 
reason lying in the fact that the student of living things is by 
nature and training an observer of concrete objects and phenomena, 
and the habit of mind which he possesses and cultivates is alien 
to that of the theoretical mathematician. But this is by no 
means the only reason; for in the kindred subject of mineralogy, 
for instance, crystals were still treated in the days of Linnaeus 
as wholly within the province of the naturalist, and were described 
by him after the simple methods in use for animals and plants: 
but as soon as Haiiy showed the application of mathematics to 


xvi] THE COMPARISON OF RELATED FORMS 721 


the description and classification of crystals, his methods were 
immediately adopted and a new science came into being. 

A large part of the neglect and suspicion of mathematical 
methods in organic morphology is due (as we have partly seen in 
our opening chapter) to an ingrained and deep-seated belief that 
even when we seem to discern a regular mathematical figure in 
an organism, the sphere, the hexagon, or the spiral which we so 
recognise merely resembles, but is never entirely explained by, 
its mathematical analogue; in short, that the details in which the. 
figure differs from its mathematical prototype are more important 
and more interesting than the features in which it agrees, and 
even that the peculiar aesthetic pleasure with which we regard 
a living thing is somehow bound up with the departure from 
mathematical regularity which it manifests as a peculiar attribute 
of life. This view seems to me to involve a misapprehension. 
There is no such essential difference between these phenomena of 
organic form and those which are manifested in portions of 
inanimate matter*. No chain hangs in a perfect catenary and no 
raindrop is a perfect sphere: and this for the simple reason that 
forces and resistances other than the main one are inevitably at 
work. The same is true of organic form, but it is for the mathe- 
matician to unravel the conflicting forces which are at work 
together. And this process of investigation may lead us on step 
by step to new phenomena, as it has done in physics, where 
sometimes a knowledge of form leads us to the interpretation of 
forces, and at other times a knowledge of the forces at work 
guides us towards a better insight into form. I would illustrate 
this by the case of the earth itself. After the fundamental advance 
had been made which taught us that the world was round, Newton 
showed that the forces at work upon it must lead to its being 
imperfectly spherical, and in the course of time its oblate spheroidal 
shape was actually verified. But now, in turn, it has been shown 
that its form is still more complicated, and the next step will be 
to seek for the forces that have deformed the oblate spheroid. 


* M. Bergson repudiates, with peculiar confidence, the application of mathe- 
matics to biology. Cf. Creative Evolution, p. 21, “Calculation touches, at most, 
certain phenomena of organic destruction. Organic creation, on the contrary, 
the evolutionary phenomena which properly constitute life, we cannot in any way 
subject to a mathematical treatment.” 


T. G. 46 


122 THE THEORY OF TRANSFORMATIONS [CH. 


The organic forms which we can define, more or less precisely, 
in mathematical terms, and afterwards proceed to explain and 
to account for in terms of force, are of many kinds, as we have 
seen; but nevertheless they are few in number compared with 
Nature’s all but infinite variety. The reason for this is not far 
to seek. The living organism represents, or occupies, a field of 
force which is never simple, and which as a rule is of immense 
complexity. And just as in the very simplest of actual cases we 
-meet with a departure from such symmetry as could only exist 
under conditions of zdeal simplicity, so do we pass quickly to 
cases where the interference of numerous, though still perhaps very 
simple, causes leads to a resultant which lies far beyond our powers 
of analysis. Nor must we forget that the biologist is much more 
exacting in his requirements, as regards form, than the physicist; 
for the latter is usually content with either an ideal or a general 
description of form, while the student of living things must needs 
be specific. The physicist or mathematician can give us perfectly 
satisfying expressions for the form of a wave, or even of a heap of 
sand; but we never ask him to define the form of any particular 


wave of the sea, nor the actual form of any monntain-peak or 
hill*. 


* In this there lies a certain justification for a saying of Minot’s, of the greater 
part of which, nevertheless, I am heartily inclined to disapprove. “‘ We biologists,”’ 
he says, “‘cannot deplore too frequently or too emphatically the great mathematical 
delusion by which men often of great if limited ability have been misled into 
becoming advocates of an erroneous conception of accuracy. The delusion is that 
no science is accurate until its results can be expressed mathematically. The 
error comes from the assumption that mathematics can express complex relations. 
Unfortunately mathematics have a very limited scope, and are based upon a few 
extremely rudimentary experiences, which we make as very little children and of 
which no adult has any recollection. The fact that from this basis men of genius 
have evolved wonderful methods of dealing with numerical relations should not 
blind us to another fact, namely, that the observational basis of mathematics is, 
psychologically speaking, very minute compared with the observational basis of 
even a single minor branch of biology....While therefore here and there the 
mathematical methods may aid us, we need a kind and degree of accuracy of which 
mathematics is absolutely incapable....With human minds constituted as they 
actually are, we cannot anticipate that there will ever be a mathematical expression 
for any organ or even a single cell, although formulae will continue to be useful 
for dealing now and then with isolated details...” (op. cit., p. 19, 1911). It were 
easy to discuss and criticise these sweeping assertions, which perhaps had their 
origin and parentage in an obiter dictum of Huxley’s, to the effect that “Mathe- 
matics is that study which knows nothing of observation, nothing of experiment, 


xvit] THE COMPARISON OF RELATED FORMS 723 


For various reasons, then, there are a vast multitude of organic 
forms which we are unable to account for, or to define, in mathe- 
matical terms; and this is not seldom the case even in forms which 
are apparently of great simplicity and regularity. The curved 
outline of a leaf, for instance, is such a case; its ovate, lanceolate, 
or cordate shape is apparently very simple, but the difficulty of 
finding for it a mathematical expression is very great indeed. 
To define the complicated outline of a fish, for instance, or of a 
vertebrate skull, we never even seek for a mathematical formula. 

But in a very large part of morphology, our essential task lies 
in the comparison of related forms rather than in the precise 
definition of each; and the deformation of a complicated figure 
may be a phenomenon easy of comprehension, though the figure 
itself have to be left unanalysed and undefined. This process 
of comparison, of recognising in one form a definite permutation 
or deformation of another, apart altogether from a precise and 
adequate understanding of the original “type” or standard of 
comparison, lies within the immediate province of mathematics, 
and finds its solution in the elementary use of a certain method 
of the mathematician. ‘This method is the Method of Co-ordinates, 
on which is based the Theory of Transformations. 

I imagine that when Descartes conceived the method of 
co-ordinates, as a generalisation from the proportional diagrams 
of the artist and the architect, and long before the immense 
possibilities of this analysis could be foreseen, he had in mind a 
very simple purpose; it was perhaps no more than to find a way 
of translating the form of a curve into numbers and into words. 
This is precisely what we do, by the method of co-ordinates, 
every time we study a statistical curve; and conversely, we 
translate numbers into form whenever we “plot a curve” to 
illustrate a table of mortality, a rate of growth, or the daily 
variation of temperature or barometric pressure. In precisely 
the same way it is possible to inscribe in a net of rectangular 
co-ordinates the outline, for instance, of a fish, and so to translate 


nothing of induction, nothing of causation” (cit. Cajori, Hist of Elem. Mathematics, 
p- 283). But Gauss called mathematics “a science of the eye’; and Sylvester 
assures us that “most, if not all, of the great ideas of modern mathematics have 
had their origin in observation” (Brit. Ass. Address, 1869, and Laws of Verse, p. 120, 
1870). 


46—2 


724 THE THEORY OF TRANSFORMATIONS [CH. 


it into a table of numbers, from which again we may at pleasure 
reconstruct the curve. 

But it is the next step in the employment of co-ordinates 
which is of special interest and use to the morphologist; and this 
step consists in the alteration, or “transformation,” of our system 
of co-ordinates and in the study of the corresponding trans- 
formation of the curve or figure inscribed in the co-ordinate 
network. . 

Let us inscribe in a system of Cartesian co-ordinates the outline 
of an organism, however complicated, or a part thereof: such as 
a fish, a crab, or a mammalian skull. We may now treat this 
complicated figure, in general terms, as a function of x, y. If we 
- submit our rectangular system to “deformation,” on simple and 
recognised lines, altering, for instance, the direction of the axes, 
the ratio of «/y, or substituting for « and y some more complicated 
expressions, then we shall obtain a new system of co-ordinates, 
whose deformation from the original type the inscribed figure 
will precisely follow. In other words, we obtain a new figure, 
which represents the old figure under strain, and is a function of 
the new co-ordinates in precisely the same way as the old figure 
was of the original co-ordinates x and y. 

The problem is closely akin to that of the cartographer who 
transfers identical data to one projection or another; and whose 
object is to secure (if it be possible) a complete correspondence, 
in each small unit of area, between the one representation and the 
other. The morphologist will not seek to draw his organic forms 
- in a new and artificial projection; but, in the converse aspect of 
the problem, he will inquire whether two different but more or 
less obviously related forms can be so analysed and interpreted 
that each may be shown to be a transformed representation of 
the other. This once demonstrated, it will be a comparatively 
easy task (in all probability) to postulate the direction and 
magnitude of the force capable of effecting the required trans- 
formation. Again, if such a simple alteration of the system of 
forces can be proved adequate to meet the case, we may find 
ourselves able to dispense with many widely current and more 
complicated hypotheses of biological causation. For it is a 
maxim in physics that an effect ought not to be ascribed to 


xvi] TH COMPARISON OF RELATED FORMS 725 


the joint operation of many causes if few are adequate to the 
production of it: Frustra fit per plura, quod fieri potest per 
pauciora. 


It is evident that by the combined action of appropriate 
forces any material form can be transformed into any other: 
just as out of a ““shapeless” mass of clay the potter or the sculptor 
models his artistic product; or just as we attribute to Nature 
herself the power to effect the gradual and successive trans- 
formation of the simplest into the most complex organism. In 
like manner it is possible, at least theoretically, to cause the outline 
of any closed curve to appear as a projection of any other what- 
soever. But we need not let these theoretical considerations 
deter us from our method of comparison of related forms. We 
shall strictly limit ourselves to cases where the transformation 
necessary to effect a comparison shall be of a simple kind, and 
where the transformed, as well as the original, co-ordinates shall 
constitute an harmonious and more or less symmetrical system. 
We should fall into deserved and inevitable confusion if, whether 
by the mathematical or any other method, we attempted to 
compare organisms separated far apart in Nature and in zoological 
classification. We are limited, not by the nature of our method, 
but by the whole nature of the case, to the comparison of 
organisms such as are manifestly related to one another and belong 
to the same zoological class. 

Our inquiry lies, in short, just within the limits which Aristotle 
himself Jaid down when, in defining a “genus,” he showed that 
(apart from those superficial characters, such as colour, which he 
called “accidents’’) the essential differences between one “ species”’ 
and another are merely differences of proportion, of relative 
magnitude, or (as he phrased it) of “excess and defect.” “Save 
only for a difference in the way of excess or defect, the parts are 
identical in the case of such animals as are of one and the same 
genus; and by ‘genus’ I mean, for instance, Bird or Fish.” 
And again: “Within the limits of the same genus, as a general 
rule, most of the parts exhibit differences...in the way of multitude 
or fewness, magnitude or parvitude, in short, in the way of excess 
or defect. For “the more’ and ‘the less’ may be represented as 


"726 THE THEORY OF TRANSFORMATIONS [CH. 


‘excess’ and ‘defect*.’’’ It is precisely this difference of relative 
magnitudes, this Aristotelian “excess and defect” in the case of 
form, which our co-ordinate method .is especially adapted to 
analyse, and to reveal and demonstrate as the main cause of what 
(again in the Aristotelian sense) we term “specific” differences. 

The applicability of our method to particular cases will depend 
upon, or be further limited by, certain practical considerations 
or qualifications. Of these the chief, and indeed the essential, 
condition is, that the form of the entire structure under investi- 
gation should be found to vary in a more or less uniform manner, 
after the fashion of an approximately homogeneous and isotropic 
body. But an imperfect isotropy, provided always that some 
“principle of continuity” run through its variations, will not 
seriously interfere with our method; it will only cause our trans- 
formed co-ordinates to be somewhat less regular and harmonious 
than are those, for instance, by which the physicist depicts the 
motions of a perfect fluid or a theoretic field of force in a uniform 
medium. 

Again, it is essential that our structyre vary in its entirety, 
or at least that “independent variants” should be relatively few. 
That independent variations occur, that localised centres of 
diminished or exaggerated growth will now and then be found, 
is not only probable but manifest; and they may even be so 
pronounced as to appear to constitute new formations altogether. 
Such independent variants as these Aristotle himself clearly 
recognised: “It happens further that some have parts that others 
have not; for instance, some [birds] have spurs and others not, 
some have crests, or combs, and others not; but, as a general 
rule, most parts and those that go to make up the bulk of the body 
are either identical with one another, or differ from one another 
in the way of contrast and of excess and defect. For ‘the more’ 
and ‘the less’ may be represented as ‘excess’ or ‘defect.’” 

If, in the evolution of a fish, for instance, it be the case that 
its several and constituent parts—head, body, and tail, or this 
fin and that fin—represent so many independent variants, then 
our co-ordinate system will at once become too complex to be 
intelligible; we shall be making not one comparison but’ several 


* Historia Animalium 1, 1. 


xvi} THE COMPARISON OF RELATED FORMS T27 


separate comparisons, and our general method will be found 
inapplicable. Now precisely this independent variability of parts 
and organs—here, there, and everywhere within the organism 
—would appear to be implicit in our ordinary accepted notions 
regarding variation; and, unless I am greatly mistaken, it is 
precisely on such a conception of the easy, frequent, and normal 
independent variability of parts that our conception of the process 
of natural selection is fundamentally based. For the morphologist, 
when comparing one organism with another, describes the 
differences between them point by point, and “character” by 
‘“character*.” If he is from time to time constrained to admit 
the existence of “correlation” between characters (as a hundred 
years ago Cuvier first showed the way), yet all the while he 
recognises this fact of correlation somewhat vaguely, as a pheno- 
menon due to causes which, except in rare instances, he can hardly 
hope to trace; and he falls readily into the habit of thinking and 
talking of evolution as though it had proceeded on the lines of his 
own descriptions, point by point, and character by charactery. 
But if, on the other hand, diverse and dissimilar fishes can be 
referred as a whole to identical functions of very different co- 
ordinate systems, this fact will of itself constitute a proof that 
variation has proceeded on definite and orderly lines, that a 
comprehensive “law of growth” has pervaded the whole structure 
in its integrity, and that some more or less simple and recognis- 
able system of forces has been at work. It will not only show 
how real and deep-seated is the phenomenon of “correlation,” 
in regard to form, but it will also demonstrate the fact that 
a correlation which had seemed too complex for analysis or 


* Cf. supra, p. 714. 

+ Cf. Osborn, H. F., On the Origin of Single Characters, as observed in fossil 
and living Animals and Plants, Amer. Nat. xt1x, pp. 193-239, 1915 (and other 
papers); ibid. p. 194, ‘Each individual is composed of a vast number of somewhat 
similar new or old characters, each character has its independent and separate 
history, each character is in a certain stage of evolution, each character is correlated 
with the other characters of the individual....The real problem has always been 
that of the origin and development of characters. Since the Origin of Species 
appeared, the terms variation and variability have always referred to single 
characters; if a species is said to be variable, we mean that a considerable number 
of the single characters or groups of characters of which it is composed are variable,” 
etc. 


728 THE THEORY OF TRANSFORMATIONS [CH. 


comprehension is, in many cases, capable of very simple graphic 
expression. This, after many trials, I believe to be in general 
the case, bearing always in mind that the occurrence of indepen- 
dent or localised variations must often be considered. 


We are dealing in this chapter with the forms of related organisms, in order 
to shew that the differences between them are as a genera! rule simple and 
symmetrical, and just such as might have been brought about by a slight and 
simple change in the system of forces to which the living and growing organism 
was exposed. Mathematically speaking, the phenomenon is identical with one 
met with by the geologist, when he finds a bed of fossils squeezed flat or other- 
wise symmetrically deformed by the pressures to which they, and the strata 
which contain them, have been subjected. In the first step towards fossilisation, 
when the body of a fish or shellfish is silted over and buried, we may take it 
that the wet sand or mud exercises, approximately, a hydrostatic pressure— 
that is to say a pressure which is un form in all directions, and by which the 
form of the buried object will not be appreciably changed. As the strata 
consolidate and accumulate, the fossil organisms which they contain will 
tend to be flattened by the vast superincumbent load, just as the stratum 
which contains them will also be compressed and will have its molecular 
arrangement more or less modified*. But the deformation due to direct 
vertical pressure in a horizontal stratum is not nearly so striking as are the 
deformations produced by the ob ique or shearing stresses to which inc’ined 
and folded strata have been exposed, and by which their various “dislocations” 
have been brought about. And espec‘ally in mountain regions, where these 
dislocations are especially numerous and complicated, the contained fossils 
are apt to be so curiously and yet so symmetrically deformed (usually by a 
simple shear) that they may easily be interpreted as so many distinct and 
separate “speciest.” A great number of described species, and here and 
there a new genus (as the genus Ellipsolithes for an obliquely deformed 
Goniatite or Nautilus) are said to rest on no other foundation f. 


If we begin by drawing a net of rectangular equidistant 
co-ordinates (about the axes x and y), we may alter or deform this 


* Cf. Sorby, Quart. Journ. Geol. Soc. (Proc.), 1879, p. 88. 

+ Cf. D’Orbigny, Alc., Cours élém. de Paléontologie, etc., 1, pp. 144-148, 1849; 
see also Sharpe, Daniel, On Slaty Cleavage, Q.J.G.S. 11, p. 74, 1847. 

t Thus Ammonites erugatus, when compressed, has been described as 4. 
planorbis: cf. Blake, J. F., Phil. Mag. (5), v1, p. 260, 1878. Wettstein has shewn 
that several species of the fish-genus Lepidopus have been based on specimens 
artificially deformed in various ways: Ueber die Fischfauna des Tertiaren 
Glarnerschiefers, Abh. Schw. Palaeont. Gesellsch. x11, 1886 (see especially pp. 
23-38, pl. 1). The whole subject, interesting as it is, has been little studied; both 
Blake and Wettstein deal with it mathematically. 


xvii] THE COMPARISON OF RELATED FORMS 129 


network in various ways, several of which are very simple indeed. 
Thus (1) we may alter the dimensions of our system, extending 
it along one or other axis, and so converting each little square 
into a corresponding and directly proportionate oblong (Fig. 353). 
It follows that any figure which we may have inscribed in the 


Y 


Fig. 354. 


original net, and which we transfer to the new, will thereby be 
deformed in strict proportion to the deformation of the entire 
configuration, being still defined by corresponding points in the 
network and being throughout in conformity with the original 
figure. For instance, a circle inscribed in the original “Cartesian” 
net will now, after extension in the y-direction, be found elongated 


730 THE THEORY OF TRANSFORMATIONS [CH. 


—— 


into an ellipse. In elementary mathematical language, for the 


original x and y we have substituted x, and cy,, and the equation 
to our original circle, 22 + y? = a?, becomes that of the ellipse, 
Le Cy 2. 

If I draw the cannon-bone of an ox (Fig. 354, A), for instance, 
within a system of rectangular co-ordinates, and then transfer 
the same drawing, point for point, to a system in which for the 
x of the original diagram we substitute x’ = 2z/3, we obtain a 
drawing (B) which is a very close approximation to the cannon- 
bone of the sheep. In other words, the main (and perhaps 
the only) difference between the two bones is simply that that of 
the sheep is elongated, along the vertical axis, as compared with 
that of the ox in the relation of 3/2. And similarly, the long 
slender cannon-bone of the giraffe (C) is referable to the same 
identical type, subject to a reduction of breadth, or increase 
of length, corresponding to 2” = 2/3. . 

(2) The second type is that where extension is not equal or 
uniform at all distances from the origin: but grows greater or 
less, as, for instance, when we stretch a tapering elastic band. 
In such cases, as I have represented it in Fig. 355, the ordinate 
increases logarithmically, and for y we substitute <”. It is obvious 
that this logarithmic extension may involve both abscissae and 
ordinates, x becoming «*, while y becomes «”. The circle in our 
original figure is now deformed into some such shape as that of 
Fig. 356. This method of deformation is a common one, and will 
often be of use to us in our comparison of organic forms. 

(3) Our third type is the “simple shear,” where the rectangular 
co-ordinates become “oblique,” their axes being inclined to one 
another at a certain angle w. Our original rectangle now becomes 
such a figure as that of Fig. 357. The system may now be 


described in terms of the oblique axes X, Y; or may be directly ~ 


referred to new rectangular co-ordinates €, 7 by the simple 
transposition x = € — n cota, y = 7 Cosec w. 

(4) Yet another important class of deformations may be 
represented by the use of radial co-ordinates, in which one set of 
lines are represented as radiating from a point or “focus,” while 
the other set are transformed into circular arcs cutting the radii 
orthogonally. These radial co-ordinates are especially applicable 


xvi] THE COMPARISON OF RELATED FORMS 731 


to cases where there exists (either within or without the figure) 
some part which is supposed to suffer no deformation; a simple 
illustration is afforded by the diagrams which illustrate the 
flexure of a beam (Fig. 358). In biology these co-ordinates will 


Fig. 358. 


be especially applicable in cases where the growing structure 
includes a “node,” or point where growth is absent or at a 
minimum; and about which node the rate of growth may be 
assumed to increase symmetrically. Precisely such a case is 
furnished us in a leaf of an ordinary dicotyledon. The leaf of a 


732 THE THEORY OF TRANSFORMATIONS [CH. 


typical monocotyledon—such as a grass or a hyacinth, for instance 
— grows continuously from its base, and exhibits no node or “ point 
of arrest.” Its sides taper off gradually from its broad base to 
its slender tip, according to some law of decrement specific to 
the plant; and any alteration in the relative velocities of longi- 
tudinal and transverse growth will merely make the leaf a little 
broader or narrower, and will effect no other conspicuous alteration 
in its contour. But if there once come into existence a node, or 
“locus of no growth,” about which we may assume the growth— — 
which in the hyacinth leaf was longitudinal and transverse—to 
take place radially and transversely to the radii, then we shall 


Fig. 359. 


at once see, in the first place, that the sloping and shghtly curved 
sides of the hyacinth leaf suffer a transformation into what we 
consider a more typical and “leaf-like” shape, the sides of the 
figure broadening out to a zone of maximum breadth and then 
drawing inwards to the pointed apex. If we now alter the ratio 
between the radial and tangential velocities of growth—in other 
words, if we increase the angles between corresponding radii— 
we pass successively through the various configurations which 
the botanist describes as the lanceolate, the ovate, and finally 
the cordate leaf. These successive changes may to some extent, 
and in appropriate cases, be traced as the individual leaf grows 


xvi] THE COMPARISON OF RELATED FORMS 733 


to maturity; but as a much more general rule, the balance 
of forces, the ratio between radial and tangential velocities of 
growth, remains so nicely and constantly balanced that the leaf 
increases in size without conspicuous modification of form. It is 
rather what we may call a long-period variation, a tendency for 
the relative velocities to alter from one generation to another, 
whose result is brought into view by this method of illustration. 
There are various corollaries to this method of describing the 
form of a leaf which may be here alluded to, for we shall not return 
again to the subject of radial co-ordinates. For instance, the 
so-called unsymmetrical leaf* of a begonia, in which one side of 
the leaf may be merely ovate while the other has a cordate outline, 
is seen to be really a case of 
unequal, and not truly asym- 
metrical, growth on either side 
of the midrib. There is nothing 
more mysterious in its conform- 
ation than, for instance, in that 
of a forked twig in which one 
limb of the fork has grown 
longer than the other. The case 
of the begonia leaf is of sufficient 
interest to deserve illustration, 
and in Fig. 360 I have outlined 
a leaf of the large Begonia dae- 
dalea. On the smaller left-hand 
side of the leaf I have taken at 
random three points, a, b, c, and 
have measured the angles, A0a, i 
etc., which the radii from the Fig. 360. Begonia daedalea. 
hilus of the leaf to these points make with the median axis. On 
the other side of the leaf I have marked the points a’, b’, c’, such 
that the radii drawn to this margin of the leaf are equal to the 
former, Oa’ to Oa, etc. Now if the two sides of the leaf are 


* Cf. Sir Thomas Browne, in The Garden of Cyrus: “But why ofttimes one 
side of the leaf is unequall unto the other, as in Hazell and Oaks, why on either 
side the master vein the lesser and derivative channels stand not directly opposite, 
nor at equall angles, respectively unto the adverse side, but those of one side do 
often exceed the other, as the Wallnut and many more, deserves another enquiry.” 


— 


734 THE THEORY OF TRANSFORMATIONS [cH. 


mathematically similar to one another, it is obvious that the 
respective angles should be in continued proportion, i.e. as AOa 
is to AOQa’, so should AOb be to AOb’. This proves to be very 
nearly the case. For I have measured the three angles on one 
side, and one on the other, and have then compared, as follows, 
the calculated with the observed values of the other two: 


AOa AOb AOc AOd’ AOl’ AOc’ 


Observed values ‘ 12° 28-5° 88° — — 157° 
Calculated  ,, — — — 21°5° 51-1° — 
Observed 3 a — — 20 52 -— 


The agreement is very close, and what discrepancy there is 
may be amply accounted for, firstly, by the slight irregularity 
of the sinuous margin of the leaf; and secondly, by the fact that 
the true axis or midrib of the leaf is not straight but slightly 


curved, and therefore that it is curvilinear and not rectilinear — 


triangles which we ought to have measured. When we under- 
stand these few points regarding the peripheral curvature of the 
leaf, it is easy to see that its principal veins approximate closely 
to a beautiful system of isogonal co-ordinates. It is also obvious 
that we can easily pass, by a process of shearing, from those cases 


where the principal veins start from the base of the leaf to those, 


as in most dicotyledons, where they arise successively from the 
midrib. 

It may sometimes happen that the node, or “point of arrest,” 
is at the upper instead of the lower end of the leaf-blade; and 
occasionally there may be a node at both ends. In the former case, 
as we have it in the daisy, the form of the leaf will be, as it were, 
inverted, the broad, more or less heart-shaped, outline appearing 
at the upper end, while below the leaf tapers gradually downwards 
to an ill-defined base. In the latter case, as in Dionaea, we obtain 
a leaf equally expanded, and similarly ovate or cordate, at both 
ends. We may notice, lastly, that the shape of a solid fruit, 
such as an apple or a cherry, is a solid of revolution, developed 
from similar curves and to be explained on the same principle. 
In the cherry we have a “ point of arrest” at the base of the berry, 
where it joins its peduncle, and about this point the fruit (in 
imaginary section) swells out into a cordate outline; while in the 


xvii] THE COMPARISON OF RELATED FORMS 735 


apple we have two sucli well-marked points of arrest, above and 
below, and about both of them the same conformation tends to 
arise. The bean and the human kidney owe their “reniform”’ 
shape to precisely the same phenomenon, namely, to the existence 
of a node or “hilus,” about which the forces of growth are radially 
and symmetrically arranged. 


Most of the transformations which we have hitherto considered 
(other than that of the simple shear) are particular cases of a 
general transformation, obtainable by the method of conjugate 
functions and equivalent to the projection of the original figure 
on a new plane. Appropriate transformations, on these general 
Imes, provide for the cases of a coaxial system where the 
Cartesian co-ordinates are replaced by coaxial circles, or a con- 
focal system in which they are replaced by confocal ellipses and 
hyperbolas. 

Yet another curious and important transformation, belonging 
to the same class, is that by which a system of straight lines 
becomes transformed into a conformal system of logarithmic 
spirals: the straight line Y—AX-=c corresponding to the 
logarithmic spiral @ — A logr = (Fig. 361). This beautiful and 
simple transformation lets us at once 
convert, for instance, the straight 
conical shell of the Pteropod or the 
Orthoceras into the logarithmic spiral 
of the Nautiloid; it involves a math- 
ematical symbolism which is but a 
shght extension of that which we 
have employed in our elementary 
treatment of the logarithmic spiral. 

These various systems of co- 
ordinates, which we have now briefly 
considered, are sometimes called “iso- 
thermal co-ordinates,’ from the fact that, when employed in 
this .particular branch of physics, they perfectly represent the 
phenomena of the conduction of heat, the contour lines of equal 
temperature appearing, under appropriate conditions, as the 
orthogonal lines of the co-ordinate system. And it follows that 


736 THE THEORY OF TRANSFORMATIONS [CH. 


the “law of growth”? which our biological analysis by means of — 


orthogonal co-ordinate systems presupposes, or at least fore- 


Ds Ha 


f 


. . . . ,- 
shadows, is one according to which the organism grows or — 


develops along stream lines, which may be defined by a suitable 
mathematical transformation. 


: 


When the system becomes no longer orthogonal, as in many ~ 


of the following illustrations—for instance, that of Orthagoriscus 


(Fig. 382),—then the transformation is no longer within the reach ~ 
of comparatively simple mathematical analysis. Such departure | 
from the typical symmetry of a “stream-line” system is, in the ~ 


first instance, sufficiently accounted for by the simple fact that 


; 


the developing organism is very far from being homogeneous and 
isotropic, or, in other words, does not behave like a perfect fluid. — 
But though under such circumstances our co-ordinate systems — 
may be no longer capable of strict mathematical analysis, they — 
will still indicate graphically the relation of the new co-ordinate 
system to the old, and conversely will furnish us with some ~ 
guidance as to the “law of growth,” or play of forces, by which — 


the transformation has been effected. 


Before we pass from this brief discussion of transformations in 
general, let us glance at one or two cases in which the forces applied 


are more or less intelligible, but the resulting transformations are, — 


from the mathematical point of view, exceedingly complicated. 
The “marbled papers” of the bookbinder are a beautiful 


illustration of visible “stream lines.” On a dishful of a sort of — 


semi-liquid gum the workman dusts a few simple lines or patches 
of colourmg matter; and then, by passing a comb through the 
liquid, he draws the colour-bands into the streaks, waves, and 
spirals which constitute the marbled pattern, and which he then 
transfers to sheets of paper laid down upon the gum. By some 
such system of shears, by the effect of unequal traction or unequal 
growth in various directions and superposed on an originally 
simple pattern, we may account for the not dissimilar marbled 
patterns which we recognise, for instance, on a large serpent’s 
skin. But it must be remarked, in the case of the marbled paper, 
that though the method of application of the forces is simple, 
yet in the aggregate the system of forces set up by the many 


xvi1] THE COMPARISON OF RELATED FORMS 737 


teeth of the comb is exceedingly complex, and its complexity is 
revealed in the complicated “diagram of forces” which constitutes 


' the pattern. 


- 


To take another and still more instructive illustration. To 
turn one circle (or sphere) into two circles would be, from the point 
of view of the mathematician, an extraordinarily difficult trans- 
formation; but, physically speaking, its achievement may be 
extremely simple. The little round gourd grows naturally, by 
its symmetrical forces of expansive growth, into a big, round, or 
somewhat oval pumpkin or melon. But the Moorish husbandman 
ties a rag round its middle, and the same forces of growth, unaltered 
save for the presence of this trammel, now expand the globular 
structure into two superposed and connected globes. And 
again, by varying the position of the encircling band, or by 
applying several such ligatures instead of one, a great variety of 
artificial forms of “gourd” may be, and actually are, produced. 
It is clear, I think, that we may account for many ordinary 
biological processes of development or transformation of form by 
the existence of trammels or lines of constraint, which limit and 
determine the action of the expansive forces of growth that would 
otherwise be uniform and symmetrical. This case has a close 
parallel in the operations of the glassblower, to which we have 
already, more than once, referred in passing*. The glassblower 
starts his operations with a tube, which he first closes at one end 
so as to form a hollow vesicle, within which his blast of air exercises 
a uniform pressure on all sides; but the spherical conformation 
which this uniform expansive force would naturally tend to 
produce is modified into all kinds of forms by the trammels or 
resistances set up as the workman lets one part or another of his 
bubble be unequally heated or cooled. It was Oliver Wendell 
Holmes who first shewed this curious parallel between the 
operations of the glassblower and those of Nature, when she starts, 
as she so often does, with a simple tubet. The alimentary canal, 


* Where gourds are common, the glass-blower is still apt to take them for a 
prototype, as the prehistoric potter also did. For instance, a tall, annulated 
Florence oil-flask is an exact but no longer a conscious imitation of a gourd which 
has been converted into a bottle in the manner described. 

{ Cf. Elsie Venner, chap. ii. 


Ven. 47 


738 THE THEORY OF TRANSFORMATIONS [CH. 


the arterial system including the heart, the central nervous 
system of the vertebrate, including the brain itself, all begin as 
simple tubular structures. And with them Nature does just 
what the glassblower does, and, we might even say, no more 
than he. For she can expand the tube here and narrow it there; 
thicken its walls or thin them; blow off a lateral offshoot or 
caecal diverticulum; bend the tube, or twist and coil it; and 
infold or crimp its walls as, so to speak, she pleases. Such a form 
as that of the human stomach is easily explained when it is 
regarded from this point of view; it is simply an ill-blown bubble, 
a bubble that has been rendered lopsided by a trammel or restraint 
along one side, such as to prevent its symmetrical expansion— 
such a trammel as is produced if the glassblower lets one side of 
his bubble get cold, and such as is actually present in the stomach 
itself in the form of a muscular band. 


We may now proceed to consider and illustrate a few permu- 
tations or transformations of organic form, out of the vast 
multitude which are equally open to this method of inquiry. 

We have already compared in a preliminary fashion the 
metacarpal or cannon-bone of the ox, the sheep, and the giraffe 
(Fig. 354); and we have seen that the essential difference in form 
between these three bones is a matter 
of relative length and breadth, such 
that, if we reduce the figures to an 
identical standard of length (or identical 
values of y), the breadth (or value of 
x) will be approximately two-thirds 
that of the ox in the case of the sheep 
and one-third that of the ox in the 
case of the giraffe. We may easily, 
for the sake of closer comparison, 
determine these ratios more accurately, 
for instance, if it be our purpose to 
compare the different racial varieties 
within the limits of a single species. 
And in such cases, by the way, as when we compare with one 
another various breeds or races of cattle or of horses, the ratios 


J 


xvit] THE COMPARISON OF RELATED FORMS 739 


of length and breadth in this particular bone are extremely 
significant*. 

If, instead of limiting ourselves to the cannon-bone, we inscribe 
the entire foot of our several Ungulates in a co-ordinate system, 
the same ratios of x that served us for the cannon-bones still give 
us a first approximation to the required comparison; but even 
in the case of such closely allied forms as the ox and the sheep 
there is evidently something wanting in the comparison. The 
reason is that the relative elongation of the several parts, or 
individual bones, has not proceeded equally or proportionately 
in all cases; in other words, that the equations for x will not 
suffice without some simultaneous modification of the values of 
y (Fig. 362). In such a case it may be found possible to satisfy 
the varying values of y by some logarithmic or other formula; 
but, even if that be possible, it will probably be somewhat difficult 
of discovery or verification in such a case as the present, owing 
to the fact that we have too few well-marked points of corre- 
spondence between the one object and the other, and that especially 
along the shaft of such long bones as the cannon-bone of the ox, 
the deer, the llama, or the giraffe there is a complete lack of easily 
recognisable corresponding points. In such a case a brief tabular 
statement of apparently corresponding values of y, or of those 
obviously corresponding values which coincide with the boundaries 
of the several bones of the foot, will, as in the following example, 
enable us to dispense with a fresh equation. 


a b c d 
y (Ox) see 0 18 27 42 100 
7 (Sheep) *.\.. On 10 19 36 100 
y (Giraffe)... 0) 5 10 24 100 


This summary of values of y’, coupled with the equations for the 


* This significance is particularly remarkable in connection with the develop- 
ment of speed, for the metacarpal region is the seat of very important leverage 
in the propulsion of the body. In the Museum of the Royal College of Surgeons 
in Edinburgh, there stand side by side the skeleton of an immense carthorse 
(celebrated for having drawn all the stones of the Bell Rock Lighthouse to the 
shore), and a beautiful skeleton of a racehorse, which (though the fact is disputed) 
there is good reason to believe is the actual skeleton of Eclipse. When I was a 
boy my grandfather used to point out to me that the cannon-bone of the little 
racer is not only relatively, but actually, longer than that of the great Clydesdale. 


47—2 


740 THE THEORY OF TRANSFORMATIONS [cH. 


value of x, will enable us, from any drawing of the ox’s foot, to 
construct a figure of that of the sheep or ay the giraffe with 
remarkable accuracy. 


That underlying the varying amounts of extension to which — 
the parts or segments of the — 


hmb have been subject there is 
a law, or principle of continuity, 
may be discerned from such a 
diagram as the above (Fig. 363), 
where the values of y in the 
case of the ox are plotted as a 
straight line, and the corre- 
sponding values for the sheep 
(extracted from the above table) 
are seen to form a more or less 
regular and even curve. This 
simple graphic result implies the 
existence of a comparatively simple equation between y and 7’. 
An elementary application of the principle of co-ordinates to 
the study of proportion, as we have here used it to illustrate the 
varying proportions of a bone, was in common use in the sixteenth 
and seventeenth centuries by artists in their study of the human 
form. The method is probably much more ancient, and may 


Fig. 363. 


Fig. 364. (After Albert Diirer.) é' 


even be classical*; it is fully described and put in practice by 
Albert Diirer in his Geometry, and especially in his Treatise on 
Proportiony. In this latter work, the manner in which the 


* Cf. Vitruvius, m1, 1. 

+ Les quatres livres d’ Albert Diirer de la proportion des parties et pourtraicts 
des corps humains, Arnheim, 1613, folio (and earlier editions). Cf. also Lavater, 
Essays on Physiognomy, U1, p. 271, 1799. 


a 


xvi] THE COMPARISON OF RELATED FORMS 741 


human figure, features, and facial expression are all transformed 
and modified by slight variations in the relative magnitude of 
the parts is admirably and copiously illustrated (Fig. 364). 

In a tapiz’s foot there is a striking difference, and yet at the 
-same time there is an obvious underlying resemblance, between 
the middle toe and either of its unsymmetrical lateral neighbours. 
Let us take the median terminal phalanx and inscribe its outline 
in a net of rectangular equidistant co-ordinates (Fig. 365, a). Let 
us then make a similar network about axes which are no longer 
at right angles, but inclined to one another at an angle of about 
50° (6). If into this new network we fill in, point for point, 
an outline precisely corresponding to our original drawing of the 
middle toe, we shall find that we have already represented the 
main features of the adjacent lateral one. We shall, however, 


perceive that our new diagram looks a little too bulky on one side, 
the inner side, of the lateral toe. If now we substitute for our 
equidistant ordinates, ordinates which get gradually closer and 
closer together as we pass towards the median side of the toe, 
then we shall obtain a diagram which differs in no essential 
respect from an actual outline copy of the lateral toe (c). In 
short, the difference between the outline of the middle toe of the 
tapir and the next lateral toe may be almost completely expressed 
by saying that if the one be represented by rectangular equidistant 
co-ordinates, the other will be represented by oblique co-ordinates, 
whose axes make an angle of 50°, and in which the abscissal 
interspaces decrease in a certain logarithmic ratio. We treated 
our original complex curve or projection of the tapir’s toe as a 
function of the form F (x, y) = 0. The figure of the tapir’s lateral 


742 THE THEORY OF TRANSFORMATIONS [cH. 


toe is a precisely identical function of the form F (e”, y,) = 0, 
where 21, y, are oblique co-ordinate axes inclined to one another 
at an angle of 50°. 


Fig. 366. (After Albert Diirer.) 


Diirer was acquainted with these oblique co-ordinates also, 
and I have copied two illustrative figures from his book*. 


In Fig. 367 I have sketched the common Copepod Oithona nana, 


Fig. 367. Otthona nana. Fig. 368. Sapphirina. 


* Tt was these very drawings of Diirer’s that gave to Peter Camper his notion 
of the “facial angle.’’ Camper’s method of comparison was the very same as ours, 
save that he only drew the axes, without filling in the network, of his coordinate 
system; he saw clearly the essential fact, that the skull varies as a whole, and that 
the “‘facial angle” is the index toa general deformation. “The great object was to 
shew that natural differences might be reduced to rules, of which the direction of 
the facial line forms the norma or canon; and that these directions and inclinations 
are always accompanied by correspondent form, size and position of the other 
parts of the cranium,” etc.; from Dr T. Cogan’s preface to Camper’s work On the 
Connexion between the Science of Anatomy and the Arts of Drawing, Painting and 
Sculpture (17687), quoted in Dr R. Hamilton’s Memoir of Camper, in Lives of 
Eminent Naturalists (Nat. Libr.), Edin. 1840. 


xvi] THE COMPARISON OF RELATED FORMS 743 


and have inscribed it in a rectangular net, with abscissae three- 
fifths the length of the ordinates. Side by side (Fig. 368) is drawn 
a very different Copepod, of the genus Sapphirina; and about 
it is drawn a network such that each co-ordinate passes (as nearly 
as possible) through points corresponding to those of the former 
figure. It will be seen that two differences are apparent. (1) The 
values of yin Fig. 368 are large in the upper part of the figure, and 
diminish rapidly towards its base. (2) The values of x are very 
large in the neighbourhood of the origin, but diminish rapidly as 
we pass towards either side, away from the median vertical axis; 
and it is probable that they do so according to a definite, but 
somewhat complicated, ratio. If, imstead of seeking for an 
actual equation, we simply tabulate our values of # and y in the 
second figure as compared with the first (just as we did in com- 
paring the feet of the Ungulates), we get the dimensions of a net 
in which, by simply projecting the figure of Ovthona, we obtain 
that of Sapphirina without further trouble, e.g.: 


x (Oithona) 0 3 6 9 12 15 — 
xv’ (Sapphirina) 0 8 10 12 13 14 —_— 
y (Otthona) 0 5 10 15 20 25 30 
y (Sapphirina) 0O 2 7 3 23 32 40 


In this manner, with a single model or type to copy from, we 
may record in very brief space the data requisite for the production 
of approximate outlines of a great number of forms. For instance 
the difference, at first sight immense, between the attenuated 
body of a Caprella and the thick-set body of a Cyamus is obviously 
little, and is probably nothing, more than a difference of relative 
magnitudes, capable of tabulation by numbers and of complete 
expression by means of rectilinear co-ordinates. 

The Crustacea afford innumerable instances of more complex 
deformations. Thus we may compare various higher Crustacea 
with one another, even in the case of such dissimilar forms as a 
lobster and a crab. It is obvious that the whole body of the 
former is elongated as compared with the latter, and that the 
crab is relatively broad in the region of the carapace, while it 
tapers off rapidly towards its attenuated and abbreviated tail. 
In a general way, the elongated rectangular system of co-ordinates 


744 THE THEORY OF TRANSFORMATIONS [CH. 


in which we may inscribe the outline of the lobster becomes a 
shortened triangle in the case of the crab. In a little more detail 
we may compare the outline of the carapace in various crabs one 
with another: and the comparison will be found easy and signifi- 
cant, even, in many cases, down to minute details, such as the 


Fig. 369. Carapaces of various crabs. 1, Geryon; 2, Corystes; 3, Seyramathia; 
4, Paralomis; 5, Lupa; 6, Chorinus. 

number and situation of the marginal spines, though these are in 
other cases subject to independent variability. 

If we choose, to begin with, such a crab as Geryon (Fig. 369, 1), 

and inscribe it in our equidistant rectangular co-ordinates, we shall 

see that we pass easily to forms more elongated in a transverse 


———————<<<— 


xvii] THE COMPARISON OF RELATED FORMS 745 


direction, such as Matuta or Lupa (5), and conversely, by 
transverse compression, to such a form as Corystes (2). In 
certain other cases the carapace conforms to a triangular dia- 
gram, more or less curvilinear, as in Fig. 4, which represents 
the genus Paralonis. Here we can easily see that the posterior 
border is transversely elongated as compared with that of Geryon, 
while at the same time the anterior part is longitudinally extended 
as compared with the posterior. A system of slightly curved and 
converging ordinates, with orthogonal and logarithmically inter- 
spaced abscissal lines, as shown in the figure, appears to satisfy 
the conditions. 

In an interesting series of cases, such as the genus Chorinus, 
or Scyramathia, and in the spider-crabs generally, we appear to 
have just the converse of this. While the carapace of these crabs 
presents a somewhat triangular form, which seems at first sight 
more or less similar to those just described, we soon see that the 
actual posterior border is now narrow instead of broad, the 
broadest part of the carapace corresponding precisely, not to 
that which is broadest in Paralomis, but to that which was broadest 
in Geryon; while the most striking difference from the latter lies 
in an antero-posterior lengthening of the forepart of the carapace, 
culminating in a great elongation of the frontal region, with its 
two spines or “horns.” The curved ordinates here converge 
posteriorly and diverge widely in front (Figs. 3 and 6), while 
the decremental interspacing of the abscissae is very marked 
indeed. 

We put our method to a severer test when we attempt to sketch 
an entire and complicated animal than when we simply compare 
corresponding parts such as the carapaces of various Malacostraca, 
or related bones as in the case of the tapir’s toes. Nevertheless, 
up to a certain point, the method stands the test very well. In 
other words, one particular mode and direction of variation is 
_ often (or even usually) so prominent and so paramount throughout 
the entire organism, that one comprehensive system of co-ordinates 
suffices to give a fair picture of the actual phenomenon. To take 
another illustration from the Crustacea, I have drawn roughly in 
Fig. 370, 1 a little amphipod of the family Phoxocephalidae 
(Harpinia sp.). Deforming the co-ordinates of the figure into the 


746 |. THE THEORY OF TRANSFORMATIONS [CH. 


curved orthogonal system in Fig. 2, we at once obtain a very fair 
representation of an allied genus, belonging to a different family 
of amphipods, namely Stegocephalus. As we proceed further from 
our type our co-ordinates will require greater deformation, and 
the resultant figure will usually be somewhat less accurate. In 
Fig. 3 I show a network, to which, if we transfer our diagram of 


Fig. 370. 1. Harpinia plumosa Kr. 2. Stegocephalus inflatus Kr. 
3. Hyperia galba. 


Harpinia or of Stegocephalus, we shall obtain a tolerable representa- 
tion of the aberrant genus Hyperia, with its narrow abdomen, 
its reduced pleural lappets, its great eyes, and its inflated head. 


The hydroid zoophytes constitute a “polymorphic” group, 
within which a vast number of species have already been dis- 
tinguished; and the labours of the systematic naturalist are 
constantly adding to the number. The specific distinctions are 
for the most part based, not upon characters directly presented 


xvi1] THE COMPARISON OF RELATED FORMS TAT 


by the living animal, but upon the form, size and arrangement 
of the little cups, or “calycles,” secreted and inhabited by the 
httle individual polypes which compose the compound organism. 
The variations, which are apparently infinite, of these conforma- 
tions are easily seen to be a question of relative magnitudes, and 
are capable of complete expression, sometimes by very simple, 
sometimes by somewhat more complex, co-ordinate networks. 
For instance, the varying shapes of the simple wineglass- 
shaped cups of the Campanularidae are at once sufficiently 
represented and compared by means of simple Cartesian co-ordi- 
nates (Fig. 371). In the two allied families of Plumulariidae and 


a b 


Fig. 371. a, Campanularia macroscyphus, Allm.; 6, Gonothyraea hyalina, 
Hineks; c, Clytia Johnstoni, Alder. 


Aglaopheniidae the calycles are set unilaterally upon a jointed 
stem, and small cup-lke structures (holding rudimentary polypes) 
are associated with the large calycles in definite number and 
position. These small calyculi are variable in number, but in the 
great majority of cases they accompany the large calycle in 
groups of three—two standing by its upper border, and one, 
which is especially variable in form and magnitude, lying at its 
base. The stem is liable to flexure and, in a high degree, to 
extension or compression; and these variations extend, often on 
an exaggerated scale, to the related calycles. As a result we find 
that we can draw various systems of curved or sinuous co-ordinates, 
which express, all but completely, the configuration of the various 


748 THE THEORY OF TRANSFORMATIONS [CH. 


hydroids which we inscribe therein (Fig. 372). The comparative 
smoothness or denticulation of the margin of the calycle, and the’ 
number of its denticles, constitutes an independent variation, and 
requires separate description; we have already seen (p. 236) that - 


d. 


Fig. 372. a, Cladocarpus crenatus, F.; 6b, Aglaophenia pluma, L.; c, A. 
rhynchocarpa, A.; d, A. cornuta, K.; e, A. ramulosa, K. ‘ 


this denticulation is in all probability due to a particular physical 
cause. a 
Among the fishes we discover a great variety of deformations, | 
some of them of a very simple kind, while others are more striking 
and more unexpected. A comparatively simple case, involving a 


Fig. 373. Arayropelecus Olfersi. Fig. 374. Sternoptyx diaphana. 


simple shear, is illustrated by Figs. 373 and 374. Fig. 373 repre- 
sents, within Cartesian co-ordinates, a certain little oceanic fish 
known as Argyropelecus Olfersi. Fig. 474 represents precisely the 
same outline, transferred to a system of oblique co-ordinates whose 


xvi] THE COMPARISON OF RELATED FORMS 749 


axes are inclined at an angle of 70°; but this is now (as far as can 
be seen on the scale of the drawing) a very good figure of an 
allied fish, assigned to a different genus, under the name of 
Sternoptyx diaphana. The deformation illustrated by this case 
of Argyropelecus is precisely analogous to the simplest and 
commonest kind of deformation to which fossils are subject (as 
we have seen on p. 553) as the result of shearing-stresses in the 
solid rock. 

Fig. 375 is an outline diagram of a typical Scaroid fish, Let us 
deform its rectilinear co-ordinates into a system of (approximately) 
coaxial circles, as in Fig. 376, and then filling into the new system, 
space by space and point by point, our former diagram of Scarus, 
we obtain a very good outline of an allied fish, belonging to a 


Fig. 375. Scarus sp. Fig. 376. Pomacanthus. 


neighbouring family, of the genus Pomacanthus. This case is all 
the more interesting, because upon the body of our Pomacanthus 
there are striking colour bands, which correspond in direction 
very closely to the lines of our new curved ordinates. In like 
manner, the still more bizarre outlines of other fishes of the same 
family of Chaetodonts will be found to correspond to very slight 
modifications of similar co-ordinates; in other words, to small 
variations in the values of the constants of the coaxial curves. 

In Figs. 377—380 I have represented another series of Acantho- 
pterygian fishes, not very distantly related to the foregoing. If 
we start this series with the figure of Polyprion, in Fig. 377, we see 
that the outlines of Pseudopriacanthus (Fig. 378) and of .Sebastes or 
Scorpaena (Fig. 379) are easily derived by substituting a system 
of triangular, or radial, co-ordinates for the rectangular ones in 


750 THE THEORY OF TRANSFORMATIONS [CH. 


which we had inscribed Polyprion. The very curious fish Anti- 
gonia capros, an oceanic relative of our own “boar-fish,” conforms 
closely to the peculiar deformation represented in Fig. 380. J 

Fig. 381 is a common, typical Diodon or porcupine-fish, and in : 
Fig. 382 I have deformed its vertical co-ordinates into a system — 


Polyprion. 


Fig. 379. Scorpaena sp. Fig. 380. Antigonia capros. 


of concentric circles, and its horizontal co-ordinates into a system 
of curves which, approximately and provisionally, are made to 
resemble a system of hyperbolas*. The old outline, transferred 

* The co-ordinate system of Fig. 382 is somewhat different from that which 
I drew and published in my former paper. It is not unlikely that further 


investigation will further simplify the comparison, and shew it to involve a still 
more symmetrical system. 


xvit] THE COMPARISON OF RELATED FORMS 751 


in its integrity to the new network, appears as a manifest 
representation of the closely allied, but very different looking, 
sunfish, Orthagoriscus mola. This is a particularly instructive 
case of deformation or transformation. It is true that, in a 
mathematical sense, it is not a perfectly satisfactory or perfectly 
regular deformation, for the system is no longer isogonal; but 


Fig. 381. Diodon. Fig. 382. Orthagoriscus. 


nevertheless, it is symmetrical to the eye, and obviously approaches 
to an isogonal system under certain conditions of friction or 
constraint. And as such it accounts, by one single integral 
transformation, for all the apparently separate and distinct 
external differences between the two fishes. It leaves the parts 
near to the origin of the system, the whole region of the head, 
the opercular orifice and the pectoral fin, practically unchanged 


152 THE THEORY OF TRANSFORMATIONS [cH. 


in form, size and position; and it shews a greater ahd greater 
apparent modification of size and form as we pass from the origin 
towards the periphery of the system. 

In a word, it is sufficient to account for the new and striking © 
contour in all its essential details, of rounded body, exaggerated 
dorsal and ventral fins, and truncated tail. In like manner, and 
using precisely the same co-ordinate networks, it appears to me 
possible to shew the relations, almost bone for bone, of the skeletons 
of the two fishes; in other words, to reconstruct the skeleton of 
the one from our knowledge of the skeleton of the other, under 
the guidance of the same correspondence as is indicated in their 
external configuration. 


The family of the crocodiles has had a special interest for the 
evolutionist ever since Huxley pointed out that, in a degree only 
second to the horse and its ancestors, it furnishes us with a close 
and almost unbroken series of transitional forms, running down 
in continuous succession from one geological formation to another. 
I should be inclined to transpose this general statement into other 
terms, and to say that the Crocodilia constitute a case in which, 
with unusually little complication from the presence of independent 
variants, the trend of one particular mode of transformation is 
visibly manifested. If we exclude meanwhile from our comparison 
a few of the oldest of the crocodiles, such as Belodon, which differ 
more fundamentally from the rest, we shall find a long series of 
genera in which we can refer not only the changing contours of | 
the skull, but even the shape and size of the many constituent 
bones and their intervening spaces or “vacuities,” to one and the 
same simple system of transformed co-ordinates. The manner 
in which the skulls of various Crocodilians differ from one another — 
may be sufficiently illustrated by three or four examples. 

Let us take one of the typical modern crocodiles as our standard 
of form, e.g. C. porosus, and inscribe it, as in Fig. 383, a, im the 
usual Cartesian co-ordinates. By deforming the rectangular net- 
work into a triangular system, with the apex of the triangle a — 
little way in front of the snout, as in b, we pass to such a form as 
C. americanus. By an exaggeration of the same process we at 
once get an approximation to the form of one of the sharp-snouted, 


xvo] THE COMPARISON OF RELATED FORMS 753 


or longirostrine, crocodiles, such as the genus Tomistoma; and, 
in the species figured, the oblique position of the orbits, the arched 
contour of the occipital border, and certain other characters suggest 
a certain amount of curvature, such as I have represented in the 
diagram (Fig. 383, b), on the part of the horizontal co-ordinates. 
In the still more elongated skull of such a form as the Indian 
Gavial, the whole skull has undergone a great longitudinal 
extension, or, in other words, the ratio of z/y is greatly diminished ; 
and this extension is not uniform, but is at a maximum in the 
_ region of the nasal and maxillary bones. This especially elongated 
region is at the same time narrowed in an exceptional degree, and 


(2 PTH 

asa SS: 
-——— <<] 
ean 


Le. 
es 
ee 
aes 


Ee 


— 


Fig. 383. A, Crocodilus porosus. B, C. americanus. C, Notosuchus terrestris. 


its excessive narrowing is represented by a curvature, convex 
towards the median axis, on the part of the vertical ordinates. 
Let us take as a last illustration one of the Mesozoic crocodiles, 
the little Notosuchus, from the Cretaceous formation. This little 
crocodile is very different from our type in the proportions of its 
skull. The region of the snout, in front of and including the frontal 
bones, is greatly shortened; from constituting fully two-thirds of 
the whole length of the skull in Crocodilus, it now constitutes less 
than half, or, say, three-sevenths of the whole; and the whole 
skull, and especially its posterior part, is curiously compact, 
broad, and squat. The orbit is unusually large. If in the diagram 
of this skull we select a number of points obviously corresponding 


TH ee 48 


754 THE THEORY OF TRANSFORMATIONS [CH. 


to points where our rectangular co-ordinates intersect particular 
bones or other recognisable features in our typical crocodile, we 
shall easily discover that the lines joining these points in Noto- 
suchus fall into such a co-ordinate network as that which is 
represented in Fig. 383, ¢. To all intents and purposes, then, this 
not very complex system, representing one harmonious “deforma- 
tion,” accounts for all the differences between the two figures, 
and is sufficient to enable one at any time to reconstruct a detailed 
drawing, bone for bone, of the skull of Notosuchus from the model 
furnished by the common crocodile. 

The many diverse forms of Dinosaurian reptiles, all of which 
manifest a strong family likeness underlying much superficial 


Fig. 384. Pelvis of (A) Stegosaurus; (B) Camptosaurus. 


diversity, furnish us with plentiful material for comparison by 
the method of transformations. As an instance, I have figured 
the pelvic bones of Stegosaurus and of Camptosaurus (Fig. 384, 
a, 6) to show that, when the former is taken as our Cartesian 
type, a slight curvature and an approximately logarithmic 
extension of the z-axis brings us easily to the configuration of 
the other. In the original specimen of Camptosaurus described 
by Marsh*, the anterior portion of the iliac bone is missing; and 
in Marsh’s restoration this part of the bone is drawn as though 
it came somewhat abruptly to a sharp point. In my figure I 


* Dinosaurs of North America, pl. LXxxt, ete. 1896. 


ee eee 


xvii] THE COMPARISON OF RELATED FORMS 755 


have completed this missing part of the bone in harmony with the 
general co-ordinate network which is suggested by our comparison 
of the two entire pelves; and I venture to think that the result 
is more natural in appearance, and more likely to be correct than 
was Marsh’s conjectural restoration. It would seem, in fact, 
that there is an obvious field for the employment of the method 
of co-ordinates in this task of reproducing missing portions of a 
structure to the proper scale and in harmony with related types. 
To this subject we shall presently return. 


Fig. 385. Shoulder-girdle of Cryptocleidus. a, young; 6, adult. 


In Fig. 385, a, b, I have drawn the shoulder-girdle of Crypto- 
cleidus, a Plesiosaurian reptile, half-grown in the one case and 
full-grown in the other. The change of form during growth in 
this region of the body is very considerable, and its nature is well 
brought out by the two co-ordinate systems. In Fig. 386 I have 


Fig. 386. Shoulder-girdle of [chthyosaurus. 


drawn the shoulder-girdle of an Ichthyosaur, referring it to 
Cryptocleidus as a standard of comparison. The interclavicle, 
which is present in Ichthyosaurus, is minute and hidden in Crypto- 
cleidus; but the numerous other differences between the two 
48—2 


756 THE THEORY OF TRANSFORMATIONS [cH. 


forms, chief among which is the great elongation in [chthyosaurus 
of the two clavicles, are all seen by our diagrams to be part and 
parcel of one general and systematic deformation. 

Before we leave the group of reptiles we may glance at the 
very strangely modified skull of Pteranodon, one of the extinct 
flying reptiles, or Pterosauria. In this very curious skull the 
region of the jaws, or beak, is greatly elongated and pointed; the 
occipital bone is drawn out into an enormous backwardly-directed 
crest; the posterior part of, the lower jaw is similarly produced 
backwards; the orbit is small; and the quadrate bone is strongly 


Fig. 387. a, Skull of Dimorphodon. b, Skull of Pteranodon. 


inclined downwards and forwards. The whole skull has a con- 
figuration which stands, apparently, in the strongest possible 
contrast to that of a more normal Ornithosaurian such as 
Dimorphodon. But if we inscribe the latter in Cartesian co- 
ordinates (Fig. 387, a), and refer our Pteranodon to a system of 
oblique co-ordinates (b), in which the two co-ordinate systems of 
parallel les become each a pencil of diverging rays, we make 
manifest a correspondence which extends uniformly pate 
all parts of these very different-looking skulls. 


We have dealt so far, and for the most part we shall continue 
to deal, with our co-ordinate method as a means of comparing one 
known structure with another. But it is obvious, as I have said, 


xvi] THE COMPARISON OF RELATED FORMS T57 


that it may also be employed for drawing hypothetical structures, 
on the assumption that they have varied from a known form in 
some definite way. And this process may be especially useful, 
and will be most obviously legitimate, when we apply it to the 
particular case of representing intermediate stages between two 
forms which are actually known to exist, in other words, of recon- 
structing the transitional stages through which the course of 
1 2 3 parole | 
FERS 
Ha aN al 
US Ae | 
me ele 
a aa 7 
7a i 
(Zl a 
FES Sa eg ee 


(adie baie VOM, Sake ral WS weer 


é 


ta) 


t) 


e 


f 


g 
Y) 


Peas | ofl Seal! 
Rees Lal 


Fig. 388. Pelvis of Archaeopteryx. 


Fig. 389. Pelvis of Apatornis. 


evolution must have successively travelled if it has brought about 
the change from some ancestral type to its presumed descendant. 
Some little time ago I sent to my friend, Mr Gerhard Heilmann 
of Copenhagen, a few of my own rough co-ordinate diagrams, in- 
cluding some in which the pelves of certain ancient and primitive 
birds were compared one with another. Mr Heilmann, who is 
both a skilled draughtsman and an able morphologist, returned 
me a set of diagrams which are a vast improvement on my own, 


758 THE THEORY OF TRANSFORMATIONS [CH.. 


and which are reproduced in Figs. 388—393., Here we have, as 
extreme cases, the pelvis of Archaeopteryx, the most ancient of 
known birds, and that of Apatornis, one of the fossil “toothed” 


ee) 3 4 s 4 if 8 g 


Fig. 390. The co-ordinate systems of Figs. 388 and 389. with three 
intermediate systems interpolated, 


Ly 
Pg ea 
ier ane 
4 
a 


C7 


is 


2M 0 CAR e on Ork ie. my eaig 


Fig. 391. The first intermediate co-ordinate network, with its 


corresponding inscribed pelvis. 


birds from the North American Cretaceous formations—a bird 
shewing some resemblance to the modern terns. The pelvis of 
Archaeopteryx is taken as our type, and referred accordingly to 


xvit] THE COMPARISON OF RELATED FORMS 759 


Cartesian co-ordinates (Fig. 388); while the corresponding co- 
ordinates of the very different pelvis of Apatornis are represented 
in Fig. 389. In Fig. 390 the outlines of these two co-ordinate 
systems are superposed upon one another, and those of three 
intermediate and equidistant co-ordinate systems are interpolated 
between them. From each of these latter systems, so determined 
by direct interpolation, a complete co-ordinate diagram is drawn, 
and the corresponding outline of a pelvis is found from each of 


peeseat 


Fig. 392. The second and third intermediate co-ordinate networks, 
with their corresponding inscribed pelves. 


these systems of co-ordinates, as in Figs. 391, 392. Finally, in 
Fig. 393 the complete series is represented, beginning with the 
known pelvis of Archaeopteryx, and leading up by our three inter- 
mediate hypothetical types to the known pelvis of Apatornis. 


Among mammalian skulls I will take two illustrations only, 
one drawn from a comparison of the human skull with that of 
the higher apes, and another from the group of Perissodactyle 


760 THE THEORY OF TRANSFORMATIONS _ [cu. 


Ungulates, the group which includes the rhinoceros, the tapir, 
and the horse. 

Let us begin by choosing as our type the skull of Hyrachyus 
agrarius, Cope, from the Middle Eocene of North America, as 


Fig. 393. The pelves of Archaeopteryx and of Apatornis, with three 
transitional types interpolated between them. 


figured by Osborn in his Monograph of the Extinct Rhino- 
ceroses* (Fig. 394). 

The many other forms of primitive rhinoceros described in 
the monograph differ from Hyrachyus in various details—in the 
characters of the teeth, sometimes in the number of the toes, and 
so forth; and they also differ very considerably in the general 


* Mem. Amer. Mus. of Nat. Hist. I, To, 1898. 


xvit] THE COMPARISON OF RELATED FORMS 761 


appearance of the skull. But these differences in the conformation 
of the skull, conspicuous as they are at first sight, will be found 
easy to bring under the conception of a simple and homogeneous 
transformation, such as would result from the application of some 
not very complicated stress. For instance, the corresponding 


cl Gaetan Cokie (ounrie| SN Mae eters Miaka Aika _! 


Fig. 394. Skull of Hyrachyus agrarius. (After Osborn.) 


Fig. 395. Skull of Aceratherium tridactylum. (After Osborn.) 


co-ordinates of Aceratherium tridactylum, as shown in Fig. 395, 
indicate that the essential difference between this skull and the 
former one may be summed up by saying that the long axis of the 
skull of Aceratherjum has undergone a slight double curvature, 
while the upper parts of the skull have at the same time been 


762 THE THEORY OF TRANSFORMATIONS [CH. 


subject to a vertical expansion, or to growth in somewhat greater 
proportion than the lower parts. Precisely the same changes, 
on a somewhat greater scale, give us the skull of an existing 
rhinoceros. 

Among the species of Acerathervum, the posterior, or occipital, 
view of the skull presents specific differences which are perhaps 
more conspicuous than those furnished by the side view; and 
these differences are very strikingly brought out by the series of 
conformal transformations which F have represented in Fig. 396. 


Fig. 396. Occipital view of the skulls of various extinct rhinoceroses 
(Aceratherium spp.). (After Osborn.) 


In this case it will perhaps be noticed that the correspondence 
is not always quite accurate in small details. It could easily 
have been made much more accurate by giving a slightly sinuous 
curvature to certain of the co-ordinates. But as they stand, 
the correspondence indicated is very close, and the simplicity of 
the figures illustrates all the better the general character of the 
transformation. 

By similar and not more violent changes we pass easily to such 
allied forms as the Titanotheres (Fig. 397); and the well-known 
series of species of Titanotherium, by which Professor Osborn has 


xvit] THE COMPARISON OF RELATED FORMS 763 


illustrated the evolution of this genus, constitutes a simple and 
suitable case for the application of our method. 

But our method enables us to pass over greater gaps than these, 
and to discern the general, and to a very large extent even the 
detailed, resemblances between the skull of the rhinoceros and 
those of the tapir or the horse. From the Cartesian co-ordinates 
in which we have begun by inscribing the skull of a primitive 
rhinoceros, we pass to the tapir’s skull (Fig. 398), firstly, by con- 
verting the rectangular into a triangular network, by which we 
represent the depression of the anterior and the progressively 
increasing elevation of the posterior part of the skull; and 
secondly, by giving to the vertical ordinates a curvature such as 
to bring about a certain longitudinal compression, or condensation, 


Fig. 397. Titanotherium robustum. Fig. 398. Tapir’s skull. 


in the forepart of the skull, especially in the nasal and orbital 
regions. 

The conformation of the horse’s skull departs from that of our 
primitive Perissodactyle (that is to say our early type of rhinoceros, 
Hyrachyus) in a direction that is nearly the opposite of that taken 
by Titanothericum and by the recent species of rhinoceros. For 
we perceive, by Fig. 399, that the horizontal co-ordinates, which 
in these latter cases became transformed into curves with the 
concavity upwards, are curved, in the case of the horse, in the 
opposite direction. And the vertical ordinates, which are also 
curved, somewhat in the same fashion as in the tapir, are very 
nearly equidistant, instead of being, as in that animal, crowded 
together anteriorly. Ordinates and abscissae form an oblique 


764 THE THEORY OF TRANSFORMATIONS [CH. 


system, as is shown in the figure. In this case I have attempted 
to produce the network beyond the region which is actually 
required to include the diagram of the ‘horse’s skull, in order to 
show better the form of the general transformation, with a part 
only of which we have actually to deal. 

It is at first sight not a little surprising to find that we can pass, 
by a cognate and even simpler transformation, from our Peris- 
sodactyle skulls to that of the rabbit; but the fact that we can 


Fig. 399. Horse’s skull. 


Fig. 400. Rabbit’s skull. 


easily do so is a simple illustration of the undoubted affinity 
which exists between the Rodentia, especially the family of the 
Leporidae, and the more primitive Ungulates. For my part, I 
would go further; for I think there is strong reason to believe 
that the Perissodactyles are more closely related to the Leporidae 
than the former are to the other Ungulates, or than the Leporidae 
are to the rest of the Rodentia: Be that as it may, it is obvious 
from Fig. 400 that the rabbit’s skull conforms to a system of 


xvir] THE COMPARISON OF RELATED FORMS 765 


co-ordinates corresponding to the Cartesian co-ordinates in which 
we have inscribed the skull of Hyrachyus, with the difference, 
firstly, that the horizontal ordinates of the latter are transformed 
into equidistant curved lines, approximately arcs of circles, with 
their concavity directed downwards; and secondly, that the 
vertical ordinates are transformed into a pencil of rays approxi- 
mately orthogonal to the circular arcs. In short, the configuration 
of the rabbit’s skull is derived from that of our primitive rhinoceros 
by the unexpectedly simple process of submitting the latter to a 


Pm aa ee Sa SAT T6 SIO 
t 


5A 


pe NSS wor Ree 


Fig. 401. A, outline diagram of the Cartesian co-ordinates of the skull of Hyra- 
cotherium or Eohippus, as shewn in Fig. 402, A. H, outline of the corresponding 
projection of the horse’s skull. B-G, intermediate, or interpolated, outlines. 


strong and uniform flexure in the downward direction (cf. Fig. 358, 
p. 731). In the case of the rabbit the configuration of the 
individual bones does not conform quite so well to the general 
transformation as it does when we are comparing the several 
Perissodactyles one with another; and the chief departures 
from conformity will be found in the size of the orbit and in the 
outline of the immediately surrounding bones. The simple fact 
is that the relatively enormous eye of the rabbit constitutes an 
independent variation, which cannot be brought into the general 
and fundamental transformation, but must be dealt with 


is UE 7 =, 


FD, 


Neer 3 SO abarg 70 
a ee. Se 


Fig. 402. A, skull of Hyracotherium, from the Eocene. after W. B. Scott; H, skull 
of horse, represented as a co-ordinate transformation of that of Hyracotherium, 


ee a oh Cet UN an? 
Saas ee ae 


fees eat) Eee 
ra 
(peo Ns SeGerges eo Wiles 


and to the same scale of magnitude; B-G, various artificial or imaginary 
types, reconstructed as intermediate stages between A and H; WM, skull of 
Mesohippus, from the Oligocene, after Scott, for comparison with C; P, skull 
of Protohippus, from the Miocene, after Cope. for comparison with H; Pp, 
lower jaw of Protohippus placidus (after Matthew and Gidley), for comparison 
with F; Mi, Miohippus (after Osborn), Pa, Parahippus (after Peterson), 
shewing resemblance, but less perfect agreement, with C and D. 


768 THE THEORY OF TRANSFORMATIONS [CH. 


separately. The enlargement of the eye, like the modification in 
form and number of the teeth, is a separate phenomenon, which 
supplements but in no way contradicts our general comparison of 
the skulls taken in their entirety. 


Before we leave the Perissodactyla and their allies, let us look 
a little more closely into the case of the horse and its immediate 
relations or ancestors, doing so with the help of a set of diagrams 


which I again owe to Mr Gerard Heilmann*. Here we start afresh,’ 


with the skull (Fig. 402, 4) of Hyracotherium (or Eohippus), 
inscribed in a simple Cartesian network. At the other end of the 


series (1) is a skull of Equus, in its own corresponding network ; | 


and the intermediate stages (B—G) are all drawn by direct and 
simple interpolation, as in Mr Heilmann’s former series of drawings 
of Archaeop'eryx and Apatornis. In this present case, the relative 
magnitudes are shewn, as well as the forms, of the several skulls. 
Alongside of these reconstructed diagrams, are set figures of 
certain extinct “horses”’ (Equidae or Palaeotheriidae), and in 


i ee ee eee 


two cases, viz. Mesohippus and Protohippus (M, P), it will be . 


seen that the actual fossil skull comcides in the most perfect 


fashion with one of the hypothetical forms or stages which our — 


method shews to be implicitly involved in the transition from 
Hyracothervum to Equus. In a third case, that of Parahippus 
(Pa), the correspondence (as Mr Heilmann points out) is by no 
means exact. The outline of this skull comes nearest to that of 
the hypothetical transition stage D, but the “fit” is now a bad 
one; for the skull of Parahippus is evidently a longer, straighter 
and narrower skull, and differs in other minor characters besides. 
In short, though some writers have placed Parahippus in the 
direct line of descent between Equus and, Hohippus, we see at 
once that there is no place for it there, and that it must, accord- 
ingly, represent a somewhat divergent branch or offshoot of the 
Equidaet. It may be noticed, especially in the case of Pro ohippus 
* These and also other coordinate diagrams will be found in Mr G. Heilmann’s 
book Fuglenes Afstamning, 398 pp., Copenhagen, 1916; see especially pp. 368-380. 
+ Cf. Zittel, Grundziige d. Palaeontologie, p. 463, 1911. 


t Cf. W. B. Scott (Amer. Journ. of Science, xivim, pp. 335-374, 1894), “We 
find that any mammalian series at all complete, such as that of the horses, is 


remarkably continuous, and that the progress of discovery is steadily fillmg up — 


xvi] THE COMPARISON OF RELATED FORMS 769 


(P), that the configuration of the angle of the jaw does not tally 
quite so accurately with that of our hypothetical diagrams as do 
other parts of the skull. As a matter of fact, this region is 
somewhat variable, in different species of a genus, and even in 
different individuals of the same species; in the small figure (Pp) 
of Protohippus placidus the correspondence is more exact. 

In considering this series of figures we cannot but be struck, 
not only with the regularity of the succession of “transformations,” 
but also with the slight and inconsiderable differences which 
separate the known and recorded stages, and even the two extremes 
of the whole series. These differences are no greater (save in 
regard to actual magnitude) than those between one human skull 
and another, at least if we take into account the older or remoter 


B Cc Se 


Fig. 403. Human scapulae (after Dwight), A, Caucasian; B, Negro; 
C, North American Indian (from Kentucky Mountains). 


races; and they are again no greater, but if anything less, than 
the range of variation, racial and individual, in certain other 
human bones, for instance the scapula*. 

The variability of this latter bone is great, but it is neither 


what few gaps remain. So closely do successive stages follow upon one another 
that it is sometimes extremely difficult to arrange them all in order, and to 
distinguish clearly those members which belong in the main line of descent, and 
those which represent incipient branches. Some phylogenies actually suffer from 
an embarrassment of riches.” 

* Cf. Dwight, T., The Range of Variation of the Human Scapula, Amer. Nat. 
XXI, pp. 627-638, 1887. Cf. also Turner, Challenger Rep. XLvIr, on Human Skele- 
tons, p. 86, 1886: “I gather both from my own measurements, and those of other 
observers, that the range of variation in the relative length and breadth of the 
scapula is very considerable in the same race, so that it needs a large number of 
bones to enable one to obtain an accurate idea of the mean of the race.” 


mT! Gs 49 - 


770 THE THEORY OF TRANSFORMATIONS .- [cu. 


surprising nor peculiar; for it is inked with all the considerations 
of mechanical efficiency and functional modification which we 
dealt with in our last chapter. The scapula occupies, as it were, 
a focus in a very-important field of force; and the lines of force 
converging on it will be very greatly modified by the varying 
development of the muscles over a large area of the body and of 
the uses to which they are habitually put. 


d 


Fig. 405. Co-ordinates of chimpanzee’s skull, as a projection of 
the Cartesian co-ordinates of Fig. 404. 


Let us now inscribe in our Cartesian co-ordinates the outline 
of a human skull (Fig. 404), for the purpose of comparing it with 
the skulls of some of the higher apes. We know beforehand that 
the main differences between the human and the simian types 
depend upon the enlargement or expansion of the brain and 
braincase in man, and the relative diminution or enfeeblement of 
his jaws. Together with these changes, the “facial angle” 
increases from an oblique angle to nearly a right angle in man, 


xvi] THE COMPARISON OF RELATED FORMS 771 


and the configuration of every constituent bone of the face and 
skull undergoes an alteration. We do not know to begin with, 
and we are not shewn by the ordinary methods of comparison, 
how far these various changes form part of one harmonious and 
congruent transformation, or whether we are to look, for instance, 
upon the changes undergone by the frontal, the occipital, the 
maxillary, and the mandibular regions as a congeries of separate 
modifications or independent variants. But as soon as we have 
marked out a number of points in the gorilla’s or chimpanzee’s 
skull, corresponding with those which our co-ordinate network 
intersected in the human skull, we find that these corresponding 
points may be at once linked up by smoothly curved lines of 
intersection, which form a new system of co-ordinates and con- 
stitute a simple “projection” of our human skull. The network 


Fig. 406. Skull of chimpanzee. Fig. 407. Skull of baboon. 


represented in Fig. 405 constitutes such a projection of the human 
skull on what we may call, figuratively speaking, the “plane” of 
the chimpanzee; and the full diagram in Fig. 406 demonstrates 
the correspondence. In Fig. 407 I have shewn the similar de- 
formation in the case of a baboon, and it is obvious that the 
transformation is of precisely the same order, and differs only in 
an increased intensity or degree of deformation. 

In both dimensions, as we pass from above downwards and 
from behind forwards, the corresponding areas of the network 
are seen to increase in a gradual and approximately logarithmic 
order in the lower as compared with the higher type of skull; 
and, in short, it becomes at once manifest that the modifications 
of jaws, braincase, and the regions between are all portions of one 
continuous and integral process. It is of course easy to draw the 


49—2 


172 THE THEORY OF TRANSFORMATIONS [CH. 


inverse diagrams, by which the Cartesian co-ordinates of the ape 
are transformed into curvilinear and non-equidistant co-ordinates 
in man. 

From this comparison of the gorilla’s or chimpanzee’s with 
the human skull we realise that an inherent weakness underlies 
.the anthropologist’s method of comparing skulls by reference to 
a small number of axes. The most important of these are the 


“facial” and “basicranial” axes, which include between them the- 


“facial angle.” But it is, in the first place, evident that these 
axes are merely the principal axes of a system of co-ordinates, 
and that their restricted and isolated use neglects all that can be 
learned from the filling in of the rest of the co-ordinate network. 
And, in the second place, the “facial axis,” for imstance, as 
ordinarily used in the anthropological comparison of one human 
skull with another, or of the human skull with the gorilla’s, is 
in all cases treated as a straight lme; but our mvestigation has 
shewn that rectiliear axes only meet the case in the simplest 
and most closely related transformations; and that, for instance, 
in the anthropoid skull no rectilinear axis is homologous with a 


rectilinear axis in a man’s skull, but what is a straight lme in the 


one has become a certain definite curve in the other. 

Mr Heilmann tells me that he has tried, but without success, 
to obtain a transitional series between the human skull and some 
prehuman, anthropoid type, which series (as in the case of the 
Equidae) should be found to contain other known types in direct 
linear sequence. It appears impossible, however, to obtain such a 
series, or to pass by successive and continuous gradations through 
such forms as Mesopithecus, Pithecanthropus, Homo neander- 
thalensis, and the lower or higher races of modern man. The 
failure is not the fault of our method. It merely indicates that 
no one straight line of descent, or of consecutive transformation, 
exists; but on the contrary, that among human and anthropoid 
types, recent and extinct, we have to do with a complex problem 
of divergent, rather than of continuous, variation. And in like 
manner, easy as it is to correlate the baboon’s and chimpanzee’s 
skulls severally with that of man, and easy as it is to see that the 
chimpanzee’s skull is much nearer to the human type than is the 
baboon’s, it is also not difficult to perceive that the series is not, 


ee 


xvii] THE COMPARISON OF RELATED FORMS 773 


strictly speaking, continuous, and that neither of our two apes 
lies precisely on the same direct line or sequence of deforma- 
tion by which we may hypothetically connect the other with 
man. 

As a final illustration I have drawn the outline of a dog’s 
skull (Fig. 408), and inscribed it in a network comparable with 
the Cartesian network of the human skull in Fig. 404. Here we 
attempt to bridge over a wider gulf than we have crossed in any 
of our former comparisons. But, nevertheless, it is obvious that 
our method still holds good, in spite of the fact that there are 
various specific differences, such as the open or closed orbit, etc., 
which have to be separately described and accounted for. We 
see that the chief essential differences in plan between the dog’s 
skull and the man’s he in the fact that, relatively speaking, the 


Fig. 408. Skull of dog, compared with the human skull of Fig. 404. 


former tapers away in front, a triangular taking the place of a 
rectangular conformation; secondly, that, coincident with the 
tapering off, there is a progressive elongation, or pulling out, of 
the whole forepart of the skull; and lastly, as a minor difference, 
that the straight vertical ordinates of the human skull become 
curved, with their convexity directed forwards, in the dog. While 
the net result is that in the dog, just as in the chimpanzee, the 
brain-pan is smaller and the jaws are larger than in man, it is 
now conspicuously evident that the co-ordinate network of the 
ape is by no means intermediate between those which fit the other 
two. The mode of deformation is on different lines; and, while 
it may be correct to say that the chimpanzee and the baboon are 
more brute-like, it would be by no means accurate to assert that 
they are more dog-like, than man. . 


174 THE THEORY OF TRANSFORMATIONS [CH. 


In this brief account of co-ordinate transformations and of 
their morphological utility I have dealt with plane co-ordinates 
only, and have made no mention of the less elementary subject 
of co-ordinates in three-dimensional space. In theory there is 
no difficulty whatsoever in such an extension of our method; it 
is just as easy to refer the form of our fish or of our skull to the 
rectangular co-ordinates x, y, 2, or to the polar co-ordinates 
E, n, ¢, as it is to refer their plane projections to the two axes to 
which our investigation has been confined. And that it would 
be advantageous to do so goes without saying; for it is the shape 


— 


of the solid object, not that of the mere drawing of the object, 


that we want to understand; and already we have found some 
of our easy problems in solid geometry leading us (as in the case 
of the form of the bivalve and even of the univalve shell) quickly 
in the direction of co-ordinate analysis and the theory of conformal 
transformations. But this extended theme I have not attempted 
to pursue, and it must be left to other times, and to other hands. 
Nevertheless, let us glance for a moment at the sort of simple 
cases, the simplest possible cases, with which such an investigation 
might begin; and we have found our plane co-ordinate systems 
so easily and effectively applicable to certain fishes that we may 
seek among them for our first and tentative introduction to the 
three-dimensional field. 

It is obvious enough that the same method of description and 
analysis which we have applied to one plane, we may apply to 
another: drawing by observation, and by a process of trial and 
error, our various cross-sections and the co-ordinate systems 
which seem best to correspond. But the new and important 
problem which now emerges is to correlate the deformation or 
transformation which we discover in one plane with that which 
we have observed in another: and at length, perhaps, after 
grasping the general principles of such correlation, to forecast 
approximately what is likely to take place in the other two planes 
of reference when we are acquainted with one, that is to say, to 
determine the values along one axis in terms of the other two. 

Let us imagine a common “round”? fish, and a common “flat” 


fish, such as a haddock and a plaice. These two fishes are not as © 


nicely adapted for comparison by means of plane co-ordinates as 


xvil] THE COMPARISON OF RELATED FORMS T75 


some which we have studied, owing to the presence of essentially 
unimportant, but yet conspicuous differences in the position of 
the eyes, or in the number of the fins,—that is to say in the manner 
in which the continuous dorsal fin of the plaice appears in the 
haddock to be cut or scolloped into a number of separate fins. 
But speaking broadly, and apart from’ such minor differences as 
these, it is manifest that the chief factor in the case (so far as we 
at present see) is simply the broadening out of the plaice’s body, 
as compared with the haddock’s, in the dorso-ventral direction, 
that is to say, along the y axis; in other words, the ratio z/y 
is much less, (and indeed little more than half as great), in the 
haddock than in the plaice. But we also recognise at once that 
while the plaice (as compared with the haddock) is expanded in 
one direction, it is also flattened, or thinned out, in the other: 
y increases, but z diminishes, relatively to x. And furthermore, 
we soon see that this is a common or even a general phenomenon. 
The high, expanded body in our Antigonia or in our sun-fish is 
at the same time flattened or compressed from side to side, in 
comparison with the related fishes which we have chosen as 
standards of reference or comparison; and conversely, such a 
fish as the skate, while it is expanded from side to side in com- 
parison with a shark or dogfish, is at the same time flattened or 
depressed in its vertical section. We proceed then, to enquire 
whether there be any simple relation of magnitude discernible 
between these twin factors of expansion and compression; and 
the very fact that the two dimensions tend to vary inversely 
already assures us that, in the general process of deformation, the 
volume is less affected than are the linear dimensions. Some years 
ago, when I was studying the length-weight co-efficient in fishes 
(of which we have already spoken in Chap. III, p. 98), that is to 
say the coefficient k in the formula W = kL’, or k= W/L, I 
was not a little surprised to find that k was all but identical in 
two such different looking fishes as our haddock and our plaice: 
thus indicating that these two fishes, little as they resemble one 
another externally (though they belong to two closely related 
families), have approximately the same volume when they are 
equal in length; or, in other words, that the extent to which the 
plaice’s body has become expanded or broadened is just about 


776 THE THEORY OF TRANSFORMATIONS [CH. 


compensated for by the extent to which it has also got flattened 
or thinned. In short, if we could permit ourselves to conceive 
of a haddock being directly transformed into a plaice, a very 
large part of the change would be simply accounted for by supposing 
the former fish to be “rolled out,” as a baker rolls a piece of dough. 
This is, as it were, an extreme case of the balancement des organes, 
or “compensation of parts.” 

Simple Cartesian co-ordinates will not suffice very well to 
compare the haddock with the plaice, for the deformation under- 
gone by the former in comparison with the latter is more on the 
lines of that by which we have compared our Antigonia with our 
Polyprion; that is to say, the expansion is greater towards the 
middle of the fish’s length, and dwindles away towards either 
end. But again simplifying our illustration to the utmost, and 
being content with a rough comparison, we may assert that, 
when haddock and plaice are brought to the same standard of 
length, we can inscribe them both (approximately) in rectangular 
co-ordinate networks, such that Y in the plaice is about twice 
as great as y in the haddock. But if the volumes of the two 
fishes be equal, this is as much as to say that xyz in the one case 
(or rather the summation of all these values) is equal to XYZ 
in the other; and therefore (since X = a, and Y = 2y), it follows 
that Z= 2/2. When we have drawn our vertical transverse 
section of the haddock (or projected that fish in the yz plane), we 
have reason accordingly to anticipate that we can draw a similar 
projection (or section) of the plaice by simply doubling the y’s 
and halving the z’s: and, very approximately, this turns out to 
be the case. The plaice is (in round numbers) just about twice 
as broad and also just about half as thick as the haddock; and 
therefore the ratio of breadth to thickness (or y to 2) is just about 
four times as great in the one case as in the other. 

It is true that this simple, or simplified, illustration carries us 
but a very little way, and only half prepares us for much greater 
complications. For instance, we have no right or reason to pre- 
sume that the equality of weights, or volumes, is a common, 
much less a general rule. And again, in all cases of more complex 
deformation, such as that by which we have compared Diodon 
with the sunfish, we must be prepared for very much more 


xvi} THE COMPARISON OF RELATED FORMS CUE 


recondite methods of comparison and analysis, leading doubtless to 
very much more complicated results. In this last case, of Diodon 
and the sunfish, we have seen that the vertical expansion of the 
latter as compared with the former fish, increases rapidly as we 
go backwards towards the tail; but we can by no means say that 
the lateral compression increases in like proportion. If anything, 
it would seem that the said expansion and compression tend to 
vary inversely; for the Diodon is very thick in front and greatly 
thinned away behind, while the flattened sunfish is more nearly 
of the same thickness all the way along. Interesting as the whole 
subject is we must meanwhile leave it alone; recognising, however, 
that if the difficulties of description and representation could be 
overcome, it is by means of such co-ordinates in space that we 
should at last obtain an adequate and satisfying picture of the 
processes of deformation and of the directions of growth*. 


* There is a paper on the mathematical study of organic forms and organic 
processes by the learned and celebrated Gustav Theodor Fechner, which I have 
only lately read, but which would have been of no little use and help to our 
argument had I known it before. (Ueber die mathematische Behandlung organ- 
ischer Gestaiten und Processe, Berichte d. k. sachs. Gesellsch., Math.-phys. Cl., 
Leipzig, 1849, pp. 50-64.) Feéchner’s treatment is more purely mathematical 
and less physical in its scope and bearing than ours, and his paper is but a short 
one; but the conclusions to which he is led differ little from our own. Let me 
quote a single sentence which, together with its context, runs precisely on the 
lines of the discussion with which this chapter of ours began. ‘‘So ist also die 
mathematische Bestimmbarkeit im Gebiete des Organischen ganz eben so gut 
vorhanden als in dem des Unorganischen, und in letzterem eben solchen oder 
aquivalenten Beschrankungen unterworfen als in ersterem; und nur sofern die 
unorganischen Formen und das unorganische Geschehen sich einer einfacheren 
Gesetzlichkeit mehr nahern als die organischen, kann die Approximation im 
unorganischen Gebiet leichter und weiter getrieben werden als im organischen. 
Dies ware der ganze, sonach rein relative, Unterschied.”’ Here in a nutshell, in 
words written some seventy years ago, is the gist of the whole matter. 

An interesting little book of Schiaparelli’s (which I ought to have known long 
ago)—Forme organiche naturali e forme geometriche pure, Milano, Hoepli, 1898— 
has likewise come into my hands too late for discussion. 


EPILOGUE. 


In the beginning of this book I said that its scope and treat- 
ment were of so prefatory a kind that of other preface it had no 
need; and now, for the same reason, with no formal and elaborate 
conclusion do I bring it to a close. The fact that I set little store 
by certain postulates (often deemed to be fundamental). of our 
present-day biology the reader will have discovered and I have 
not endeavoured to conceal. But it is not for the sake of polemical 
argument that I have written, and the doctrines which I do not 
subscribe to I have only spoken of by the way. My task is finished 
if I have been able to shew that a certain mathematical aspect of 
morphology, to which as yet the morphologist gives little heed, is 
interwoven with his problems, complementary to his descriptive 
task, and helpful, nay essential, to his proper study and com- 
prehension of Form. Hic artem remumque repono. 

And while [ have sought to shew the naturalist how a few 
mathematical concepts and dynamical principles may help and 
guide him, I have tried to shew the mathematician a field for his 
labour,—a field which few have entered and no man has explored. 
Here may be found homely problems, such as often tax the 
highest skill of the mathematician, and reward his ingenuity all 
the more for their trivial associations and outward semblance of 
simplicity. 

That I am no skilled mathematician I have had little need to 
confess, but something of the use and beauty of mathematics I 
think I am able to understand. I know that in the study of 
material things, number, order and position are the threefold clue 
to exact knowledge; that these three, in the mathematician’s 
hands, furnish the “first outlines for a sketch of the Universe”; 


that by square and circle we are helped, like Emile Verhaeren’s 


carpenter, to conceive “Les lois indubitables et fécondes Qui sont 
la régle et la clarté du monde.” 

For the harmony of the world is made manifest in Form and 
Number, and the heart and soul and all the poetry of Natural 


= 


Ee 


EPILOGUE 1719, 


Philosophy are embodied in the concept of mathematical beauty. 
A greater than Verhaeren had this in mind when he told of “ the 
golden compasses, prepared In God’s eternal store.” A greater 
than Milton had magnified the theme and glorified Him ‘“‘ who 
sitteth upon the circle of the earth,” saying: He measureth the 
waters in the hollow of his hand, he meteth out the heavens with 
his span, he comprehendeth the dust of the earth in a measure. 

Moreover the perfection of mathematical beauty is such (as 
Maclaurin learned of the bee), that whatsoever is most beautiful 
and regular is also found to be most useful and excellent. 

The living and the dead, things animate and inanimate, we 
dwellers in the world and this world wherein we dwell,—zavra 
ya pav Ta yuyywoKoueva,—are bound alike by physical and 
mathematical law. “Conterminous with space and coeval with 
time is the kingdom of Mathematics;'’ within this range her 
dominion is supreme; otherwise than according to her order 
nothing can exist, and nothing takes place in contradiction to her 
laws.” So said, some forty years ago, a certain mathematician; 
and Philolaus the Pythagorean had said much the same. 

But with no less love and insight has the science of Form and 
Number been appraised in our own day and generation by a very 
great Naturalist indeed:—by that old man eloquent, that wise 
student and pupil of the ant and the bee, who died but yesterday, 
and who in his all but saecular life tasted of the firstfruits of 
immortality; who curiously conjoined the wisdom of antiquity 
with the learning of to-day; whose Provencal verse seems set to 
Dorian music; in whose plainest words is a sound as of bees’ 
industrious murmur; and who, being of the same blood and 
marrow with Plato and Pythagoras, saw in Number “la clef de la 
voute,” and found in it “le comment et le pourquoi des choses.” 


INDEX. 


Abbe’s diffraction plates 323 

Abel, O. 706 

Abonyi, A. 127 

Acantharia, spicules of 458 

Acanthometridae 462 

Acceleration 64 

Aceratherium 761 

Achlya 244 

Acromegaly 135 

Actinomma 469 

Actinomyxidia 452 

Actinophrys 165, 197, 264, 298 

Actinosphaerium 197, 266, 298, 468 

Adams, J. C. 663 

Adaptation 670 

Addison, Joseph, 671 

Adiantum 408 

Adsorption 192, 208, 241, 277, 357; 
orientirte 440, 590; pseudo 282 

Agelutination 201 

Aglaophenia 748 

Airy, H. 636 

Albumin molecule 41 

Aleyonaria 387, 413, 424, 459 

Alexeieff, A. 157, 165 

Allmann, W. 643 

Alpheus, claws of 150 

Alpine plants 124 

Altmann’s granules 285 

Alveolar meshwork 170 

Ammonites 526, 530, 537, 539, 550, 
552, 576, 583, 584, 728 ‘ 

Amoeba 12, 165, 209, 212, 245, 255, 
288, 463, 605 

Amphidiscs 440 

Amphioxus 311 

Ampullaria 560 

Anabaena 300 

Anaxagoras 8 

Ancvloceras 550 

Andrews, G. F. 

Anhydrite 433 

Anikin, W. P. 130 

Anisonema 126 

Anisotropy 241, 357 

Anomia 565, 567 

Antelopes, horns of 614, 671 

Antheridia 303, 403, 405, 409 


164; C. W. 716 


Anthoceros, spore of 397 

Anthogorgia, spicules of 413 

Anthropometry 51 

Anticiine 360 

Antigonia 750, 775 

Antlers 628 

Apatornis 757 

Apocynum, pollen of 396 

Aptychus 576 

Arachnoidiscus 387 

Arachnophyllum 325 

Arcella 323 

Arcestes 539, 540 

Archaeopteryx 757 

Archimedes 580; spiral of 503, 524, 
552 

Argali, horns of 617 

Argiope 561 

Argonauta 546, 561 

Argus pheasant 431, 631 

Argyropelecus 748 

Aristotle 3, 4, 5, 8, 15, 138, 149, 158, 
509, 653, 714, 725, 726 

Arizona trees 121 

Arrhenius, Sv. 28, 48, 171 

Artemia 127 

Artemis 561 

Ascaris megalocephala 180, 195 

Aschemonella 255 

Assheton, R. 344 

Asterina 342 

Asteroides 423 

Asterolampra 386 

Asters 167, 174 

Asthenosoma 664 

Astrorhiza 255, 463, 587, 607 

Astrosclera 436 

Asymmetric substances 416 

Asymmetry 241 

Atrypa 569 

Auerbach, F. 9 

Aulacantha 460. 

Aulastrum 471 

Aulonia 468 

Auricular height 93 

Autocatalysis 131 

Auximones 135 

Awerinzew, 8. 589 


INDEX 781 


Babak, E. 32 

Babirussa, teeth of 634 
Baboon, skull of 771 

Bacillus 39; B. ramosus 133 


Bacon, Lord 4, 5, 51, 53, 131, 656, 


716 
Bacteria 245, 250 
Bact ko be vonpo,, 005-072, 15 
Balancement 714, 776 
Balfour, F. M. 57, 348 
Baltzer, Fr. 327 
Bamboo, growth of 77 
Barclay, J. 334 
Barfurth, D. 85 
Barlow, W. 202 
Barratt, J. O. W. 285 
Bartholinus, E. 329 
Bashforth, Fr. 663 
Bast-fibres, strength of 679 
Baster, Job 138 
Bateson, W. 104, 431 
Bather, F. A. 578 
Batsch, A. J. G. K. 606 
Baudrimont, A., and St Ange 124 
Baumann and Roos 136 
Bayliss, W. M. 135, 277 
Beads or globules 234 
Beak, shape of 632 
Beal, W. J. 643 
Beam, loaded 674 
Bee’s cell 327, 779 
Begonia 412, 733 
Beisa antelope, horns of 616, 621 
Bellerophon 550 
Bénard, H. 259, 319, 448, 590 
Bending moments 19, 677, 696 
Beneden, Ed. van 153, 170, 198 
. Bergson, H. 7, 103, 251, 611, 721 
Bernard, Claude 2, 13, 127 
Bernoulli, ieee 580; John 30, 54 
Berthold, G. 
Sol, aos Ps 372, 399 
Bethe, A. 276 
Bialaszewicz, K. 114, 125 
Biedermann, W. 431 
Bilharzia, egg of 656 
Binuclearity 286 
Biocrystallisation 454 
Biogenetisches Grundgesetz 608 
Biometrics 78 
Bird, flight of 24; form of 673 
Bisection of solids 352, ete. 
Bishop, John 31 
Bivalve shells 561 
Bjerknes, V. 186 
Blackman, F. 
131, 132 
Blackwall, J. 234 
Blake, J. F. 536, 547, 553, 
728 


Or 


234, 298, 306, 322, 346, 


F., 108, 110, 114, 124, 


578, 583, 


Blastosphere 56, 344 

Blood-corpuscles, form of 
of 36 

Blood-vessels 665 

Boas, Fr. 79 

Bodo 230, 269 

Boerhaave, Hermann 380 

Bonanni, F. 318 

Bone, 425, 435; repair of 687; strue- 
ture of 673, 680 

Bonnet, Ch. 108, 138, 334, 635 

Borelli, J. A. 8, 27, 29, 318, 677, 690 

Bosanquet, B. 5 

Boscovich, Father R. J., S.J. 8 

Bose, J. C. 87 

Bostryx 502 

Bottazzi, F. 127 

Bottomley, J. T. 135 

Boubée, N. 529 

Bourgery, J. M. 683 

Bourne, G. C. 199 

Bourrelet, Plateau’s 
A477 

Boveri, Th. 38, 147, 170, 198 

Bowditch, H. P. 61, 79 

Bower, F. O. 406 

Bowman, J. H. 428 

Boyd, R. 61 

Boys, €. V. 233 

Brachiopods 561, 568, 577 

Bradford, 8. C. 428 

Brady, H. B. 255, 606 

Brain. growth of 89; weight of 90 

Branchipus 128, 342 ; 

Brandt, K. 459, 482 

Brauer, A. 180 

Braun, A. 636 

Bravais, L. and A. 202, 502, 636 

Bredig, G. 178 

Brewster, Sir D. 209, 337, 350, 431 

Bridge, T. W. 671 

Bridge construction 18, 691 

Brine shrimps 127 

Brooke, Sir V. 614, 624, 628, 631 


270; size 


297, 339, 446, 470, 


_ Browne, Sir T. 324, 329, 480, 650, 652, 


733 
Brownian movement 45, 279, 421 
Briicke. C. 160, 199 
Bucecinum 520, 527 
Buch, Leopold von 528, 583 
Buchner, Hans 133 
Budding 213, 399 
Bufton, on the bee’s cell 333 
Biihle, C. A. 653 
Bulimus 549, 556 
Burnet, J. 509 
Biitschli, O. 165, 170, 171, 
434, 458, 492 
Biittel-Reepen, H. von 332 
Byk, A. 419 


204, 432, 


182 


Cactus, sphaerocrystals in 434 

Cadets, growth of German 119 

Calandrini, G. L. 636 

Calcospherites 421, 434 

Callimitra 472 

Callithamnion, spore of 396 

Calman, T. W. 149 

Calyptraea 556 

Camel 703, 704 

Campanularia 237, 262, 747 : 

Campbell, D. H. 302, 397, 402 

Camper, P. 742 

Camptosaurus 754 

Cannon bone 730 

Cantilever 678, 694 

Cantor, Moritz 503 

Caprella 743 

Caprinella 567, 577 

Carapace of crabs 744 

Cardium 561 

Cariacus 629 

Carlier, E. W. 211 

Carnoy, J. B. 468 

Carpenter, W. B. 45, 422, 465 

Caryokinesis 14, 157, etc. 

Cassini, D. 329 

Cassis 559 

Catabolic products 435 

Catalytic action 130 

Catenoid 218, 223, 227, 252 

Causation 6 

Cavolinia 573 

Cayley, A. 385 

Celestite 459 

Cell-theory 197, 199 

Cells, forms of 201; sizes of 35 

Cellular pathology 200; tissue, artificial 
320 

Cenosphaera 470 

Centres of force 156, 196 

Centrosome 167, 168, 173 

Cephalopods 548, etc.; eggs of 378 

Ceratophyllum, srowth of 97 

Ceratorhinus 612 

Cerebratulus, egg of 189 

Cerianthus 125 

Cerithium 530, 557, 559 

Chabrier, J. 25 

Chabry, L. 30, 306, 415 

Chaetodont fishes 671, 749 

Chaetopterus, egg of 195 

Chamois, horns of 615 

Chapman, Abel 672 

Chara 303 

Characters, biological 196, 727 

Chevron bones 709 

Chick, hatching of 108 

Chilomonas 114 

Chladni figures 386, 475 

Chlorophyll 291 


INDEX 


Choanoflagellates 253 
Chodat, R. 78, 132 
Cholesterin 272 
Chondriosomes 285 
Chorinus 744 

Chree, C. 19 
Chromatin 153 
Chromidia 286 


Chromosomes 157, 173, 179, 181, 190, 
195 


Church, A. H. 639 

Cicero 62 

Cicinnus 502 

Cidaris 664 

Circogonia 479 

Cladocarpus 748 

Claparéde, E. R. 423 

Clathrulina 470 

Clausilia 520, 549 

Claws 149, 632 

Cleland, John 4 

Cleodora 570-575 

Climate and growth 121 

Clio 570 

Close packing 453 

Clytia 747 

Coan, C. A. 514 

Coassus 629 

Cod, otoliths of 432; skeleton of 710 

Codonella 248 

Codosiga 253 

Coe, W. R. - 189 

Coefficient of growth 153; 
perature 109 

Coelopleurus 664 

Cogan, Dr T. 742 

Cohen, A. 110 

Cohesion figures 259 

Collar-cells 253 

Colloids 162, 178, 201, 279, 412, 421, 
ete. 

Collosclerophora 436 

Collosphaera 459 

Colman, 8. 514 

Comoseris 327 

Compensation, law of 714, 776 

Conchospiral 531, 539, 594 

Conchyliometer 529 

Concretions 410, etc. 

Conjugate curves 561, 613 

Conklin, E. G. 36, 191, 310, 340, 377 

Conostats 427 

Continuous girder 700 

Contractile vacuole 165, 264 

Conus 557, 559. 560 

Cook, Sir T. A. 493, 635, 639, 650° 

Co-ordinates 723 

Corals 325, 388, 423 

Cornevin, Ch. 102 

Cornuspira 594 


of tem- 


ee ee 


ee oe 
TS ae} 


INDEX 


Correlation 78, 727 
Corystes 744 

Cotton, A. 418 

Cox; J. ; 46 

Crane-head, 682 

Crayfish, sperm-cells of, 273 
Creodonta 716 

Crepidula 36, 310, 340 
Creseis 570 

Cristellaria 515, 600 
Crocodile 704, 752 

Crocus, growth of 88 
Crookes, Sir W. 32 
Cryptocleidus 755 

Crystals 202, 250, 429, 444, 480, 601 
Ctenophora 391 

Cube, partition of 346 
Cucumis, growth of 109 
Culmann, Professor C. 682, 697 
Cultellus 564 

Curlew, eggs of 652 
Cushman, J. A. 323 
Cuvier 727 

Cuvierina 258, 570 
Cyamus 743 
Cyathophylltm 325, 391 
Cyclammina 595, 596, 602 
Cyclas 561 

Cyclostoma 554 

Cylinder 218, 227, 377 
Cymba 559 

Cyme 502 

Cypraea 547, 554, 560, 561 
Cyrtina 569 

Cyrtocerata 583 
Cystoliths 412 


Daday de Dees, E. v. 130 

Daffner, Fr. 61, 118 

Dalyell, Sir John G. 146 

Danilewsky, B. 135 

Darling, C. R. 219, 257, 664 

D’Arsonval, A. 192, 281 

Darwin, C. 4, 44, 57, 332, 431, 465, 
549, 624, 671, 714 

Dastre, A. 136 

Davenport, C. B. 107, 123, 125, 126, 211 

De Candolle, A. 108, 643; A. P. .20; 
C. 636 

Decapod Crustacea, sperm-cells of 273 

Deer, antlers of 628 

Deformation 638, 728, etc. 

Degree, differences of, 586, 725 

Delage, Yves 153 

Delaunay, C. E. 218 

Delisle 31 

Dellinger, O. P. 212 

Delphinula 557 

Delpino, F. 636 

Democritus 44 


783 


Dendy, A. 137, 436, 440, 671 
Dentalium 535, 537, 546, 555, 556, 561 
Dentine 425 

Descartes, R. 185, 723 

Des Murs, O. 653 

Devaux, H. 43 

De Vries, H. 108 

Diatoms 214, 386, 426 

Diceras 567 

Dickson, Alex. 647 

Dictyota 303, 356, 474 

Diet and growth 134 

Difflugia 463, 466 

Diffusion figures 259, 430 
Dimorphism of earwigs 105 
Dimorphodon 756 

Dinenympha 252 

Dinobryon 248 

Dinosaurs 702, 704, 754 

Diodon 751, 777 

Dionaea 734 

Diplodocus 702, 706, 710 

Disc, segmentation of a 367 
Discorbina 602 

Distigma 246 

Distribution, geographical 457, 606 
Ditrupa 586 

Dixon, A. F. 684 

Dobell, C. C. 286 

Dodecahedron 336, 478, etc. 
Doflein, F. J. 46, 267, 606 

Dog’s skull 773 

Dolium 526, 528, 530, 557, 559, 560 
Dolphin, skeleton of 709 
Donaldson, H. H. 82, 93 
Dorataspis 481 

D’Orbigny, Alc. 529, 555, 591, 728 
Douglass, A. E. 121 


Draper, J. W. 165, 264 
Dreyer, F. R. 435, 447, 455, 468, 606, 
608 


Driesch, H. 4, 35, 157, 306, 310, 312, 
377, 378, 714 

Dromia, 275 

Drops 44, 257, 587 

Du Bois-Reymond, Emil 1, 92 

Duerden, J. E. 423 

Dufour, Louis 219 

Dujardin, F. 257, 591 

Dunan 7 

Duncan, P. Martin 388 

Dupré, Athanase 279 

Durbin, Marion L. 138 

Diirer, A. 55, 740, 742 

Dutrochet, R. J. H. 212, 624 

Dwight, T. 769 

Dynamical similarity 17 


Earthworm, calcospheres in 423 
Earwigs, dimorphism in 104 


‘ 


784 


Ebner, V. von 444, 683 

Echinoderms, larval 392; spicules of 
449 

Echinus 377, 378, 664 

Eclipse, skeleton of 739 

Ectosare 281 

Eel, growth of, 85 

Efficiency, mechanical 670 

Efficient cause 6, 158, 248 

Eggs of birds 652 

Hiffel tower 20 

Fight cells, grouping of 381, etc. 

Eimer, Th. 606 

Einstein formula 47 

Elastic curve 219, 265, 271 

Elaters 489 

Electrical convection 187; stimulation 
of growth 153 

Elephant 21, 633, 703, 704 

Elk, antlers of 629, 632 

Ellipsolithes 728 

Ellis, R. Leslie 4, 329, 647; M. M. 


147, 656 
Elodea 322 
Emarginula . 556 
Emmel, V. E. 149 


Empedocles 8 

Emperor Moth 431 

Encystment 213, 283 

Engelmann, T. W. 210, 285 

Enriques, P. 4, 36, 64, 133, 134, 677 

Entelechy 4, 714 

Entosolenia 449 

Enzymes 135 

Kpeira 233 

Epicurus 47 

Epidermis 314, 370 

Epilobium, pollen of 396 

Epipolic force 212 

Equatorial plate 174 

Equiangular spiral 50, 505 

Equilibrium, figures of 227 

Equipotential lines 640 

Equisetum, spores of 290, 489 

Errera, Leo. 8, 40, 110, 111, 213, 306, 
346, 348, 426 

Erythrotrichia 358, 372, 390 

Ethmosphaera 470 

Kuastrum 214 

Kucharis 391 

Euclid 509 

Euglena 376 

Euglypha 189 

Euler, L. 3, 208, 385, 484, 690 

Eulima 559 

Eunicea, spicules of 424 

Euomphalus 557, 559 

Evelyn, John 652 

Evolution 549, 610, etc. 

Ewart, A. J. 20 


INDEX 


Fabre, J. H. 64, 779 

Facial angle 742, 770, 772 

Faraday, M. 163, 167, 428, 475 

Farmer, J. B. and Digby 190 

Fatigue, molecular 689 

Faucon, A. 88 

Favosites 325 

Fechner, G. T. 654, 777 

Fedorow, E. 8. von 338 

Fehling, H. 76, 126 

Ferns, spores of 396 

Fertilisation 193 

Fezzan-worms 127 

Fibonacci 643 

Fibrillenkonus 285 

Fick, R. 57, 683 

Fickert, C. 606 

Fidler, Prof. T. Claxton 691, 674, 

696 

Films, liquid 215, 217, 426 

Filter-passers 39 

Final cause 3, 248, 714 

Fir-cone 635, 647 

Fischel, Alfred 88 

Fischer, Alfred 40, 172; 
418; Otto 30, 699 ~ 

Fishes, forms of 748 

Fission, multiplication by 151 

Fissurella 556 

FitzGerald, G. F. 
477 

Flagellum 246, 267, 291 

Flemming, W. 170, 172, 180 

Flight 24 

Flint, Professor 673 

Fluid crystals 204, 272, 485 

Fluted pattern 260 

Fly’s cornea 324 

Fol, Hermann 168, 194 

Folliculina 249 

Foraminifera 214, 255, 415, 495, 515 

Forth Bridge 694, 699, 700 

Fossula 390 

Foster, M. 185 

Fraas, E. 716 

Frankenheim, M. L. 202 

Frazee, O. E. 153 

Frédéricq, L. 127, 130 

Free cell formation 396 

Friedenthal, H. 64 

Frisch, K. von 671 

Frog,: egg of 310, 363, 378, 382; 
growth of 93, 126 

Froth or foam 171, 205, 305, 314, 322, 


Emil 417, 


158, 281, 323, 440, 


Fucus 
Fundulus 125 
Fusulina 593, 594 
Fusus 527, 557 


7" 


INDEX : 


Gadow, H. F. 628 

Galathea 273 

Galen 3, 465, 656 

Galileo 8, 19, 28, 562, 677, 720 
Gallardo, A. 163 

Galloway, Principal 672 
Gamble, F. A. 458 
Ganglion-cells, size of 37 
Gans, R. 46 

Garden of Cyrus 324, 329 
Gastrula 344 

Gauss, K: EF. 207, 278, 723 
Gebhardt, W. 430, 683 
Gelatination, water of 203 
Generating curves and spirals 526, 

561, 615, 637, 641 

Geodeties 440, 488 
_ Geoffroy St Hilaire, Et. de 714 
Geotropism 211 

Gerassimow, J. J. 35 

Gerdy, P. N. 491 

Geryon 744 

Gestaltungskraft 485 

Giard, A. 156 

Gilmore, C. W. 707 

Giraffe 705, 730, 738 
Girardia 321, 408 

Glaisher, J. 250 
Glassblowing 238, 737 

Gley, E. 135, 136 
Globigerina 214, 234, 440, 495, 589, 

602, 604, 606 

Gnomon 509, 515, 591 

‘Goat, horns of 613 

Goat moth, wings of 430 ; 
Goebel, K. 321, 397, 408 
Goethe 20, 38, 199, 714, 719 
Golden Mean 511, 643, 649 
Goldschmidt, R. 286 
Goniatites 550, 728 — 
Gonothyraea 747 

Goodsir, John 156, 196, 580 
Gottlieb, H. 699 

Gourd, form of 737 

Grabau, A. H. 531, 539, 550 
Graham, Thomas 162, 201, 203 
Grant, Kerr 259 - 
Grantia 445 

Graphic statics 682 
Gravitation 12, 32 

Gray, J. 188 

Greenhill, Sir A. G. 19 
Gregory. D. F. 330, 675 
Greville, R. K. 386 

Gromia 234, 257 

Gruber, A. 165 

Gryphaea 546, 576, 577 
Guard-cells 394 

Gudernatsch, J. F. 136 
Guillemot, egg of 652 


T. G. 


ae | 
CO 
or 


Gulliver, G. 36 
Giinther, F. C. 653, 654 
Gurwitsch, A. 285 


Hacker, V. 458 

Haddock 774 

Haeckel, E. 199, 445, 454, 455, 457, 
467, 480, 481 

Hair, pigmentation of 430 

Hales, Stephen 36, 59, 95 669 

Haliotis 514, 527, 546, 547, 554, 555, 
557, 561 

Hall, Cs EK. * 119 

Haller, A. von 2, 54, 56, 59, 64, 68 

Hardesty, Irving 37 

Hardy, W. B. 160, 162, 172, 187, 287 

Harle, N. 28 

Harmozones 135 

Harpa 526, 528, 559 

Harper, R. A. 283 

Harpinia 746 

Harting, P. 282, 420, 426, 434 

Hartog, M. 163, 327 

Harvey, EK. N. and H. W. 187 

Hatam Ss. 132, 135 

Hatchett, C. 420 

Hatschek, B. 180 

Haughton, Rev. 8. 334, 666 

Haiiy, R. J. 720 

iHayenO nb.) 107 

Haycraft, J. B. 211, 690 

Head, length of 93 

Heart, growth of 89; muscles of 490 

Heath, Sir T. 511 

Hegel, G. W. F. 4 

Hegler 680, 688 

Heidenhain, M. 170, 212 

Heilmann, Gerhard 757, 768, 772 

Helicoid 230; cyme 502, 605 

Helicometer 529 

Helicostyla 557 

Heliolites 326 

Heliozoa 264, 460 

Helix 528, 557 

Helmholtz, H. von 2, 9, 25 

Henderson, W. P. 323 

Henslow, G. 636 

Heredity 158, 286, 715 

Hermann, F. 170 

Hero of Alexandria 509 

Heron-Allen, E. 257, 415, 465 

Herpetomonas 268 

Hertwig, O. 56, 114, 153, 199, 310; 
R. 170, 285 

Hertzog, R. O. 109 

Hess, W. 666, 668 

Heteronymous horns 619 

Heterophyllia 388 

Hexactinellids 429, 452, 453 

Hexagonalsymmetry 319, 323,471,513 


50 


786 


Hickson, S. J. 424 

Hippopus 561 

His, W. 55, 56, 74, 75 

Hobbes, Thomas 159 

Hober, R. 1, 126, 130, 172 

Hodograph 516 

Hoffmann, C. 628 

Hofmeister, F. 41; W. 87, 210, 234, 
304, 306, 636, 639 

Holland, W. J. 707 

Holmes, O. W. 62, 737 

Holothuroid spicules 440, 451 

Homonymous horns 619 

Homoplasy 251 

Hooke, Robert 205 

Hop, growth of 118; 

Horace 44 

Hormones 135 

Horns 612 

Horse 694, 701, 703, 764 

Houssay, F. 21 

Huber, P. 332 

Huia bird 633 

Humboldt, A. von 127 

Hume, David 6 

Hunter, John 667, 669, 713, 715 

Huxley, Algal 423, 722, 752 

Hyacinth 322, 394 

Hyalaea 571-577 

Hyalonema 442 

Hyatt, A. 548 

Hyde, Ida H. 125, 163, 184, 188 

Hydra 252; egg of 164 

Hydractinia 342 

Hydraulics 669 

Hydrocharis 234 

Hyperia 746 

Hyrachyus 760, 765 

Hyracotherium 766, 768 


stem of, 627 


Ibex 617 

Ice, structure of 428 
Ichthyosaurus 755 
Icosahedron 478 
Iguanodon 706, 708 
Inachus, sperm-cells of 273 
Infusoria 246, 489 
Intussusception 202 

Inulin 432 

Invagination 56, 344 
lodine 136 

Irvine, Robert 414, 434. 
Isocardia 561, 577 | 
Isoperimetrical problems 208, 346 
[sotonic solutions 130, 274 
Iterson, G. van 595 


Jackson, C. M. 7 
Jamin, J. C. 418 
Janet, Paul 5, 18, 673 


5, 88, 106 


INDEX 


Japp, F. R. 417 
Jellett, J. H. 1 
Jenkin, C. F. 444 
Jenkinson, J. W. 
Jennings, H. 8. 212, 
424 
Jensen, P. 211 
Johnson, Dr 8. 62 
Joly, John 9, 63 
Jost; Le TOME ' 


94, 114, 170 
492; Vaughan 


Juncus, pith of 335 


Jungermannia 404 


Kangaroo 705, 706, 709 

Kanitz, Al. 109 

Kant, Immanuel 1, 3, 714 

Kappers, C. U. A. 566 

Kellicott, W. E. 91 

Kelvin, Lord 9, 49, 188, 202, 336, 453 
Kepler 328, 480, 486, 643, 650 
Kienitz-Gerloff, F. 404, 408 

Kirby and Spence 28, 30, 127 


Kirehner, A. 683 
Kirkpatrick, R. 437 
Klebs, G. 306 
Kny, L. 680 

Koch, G. von 423 
Koenig, Sar a! 330 
Kofoid, C. A. 268 


Kolliker, A. von 413 
Kollmann, M. 170 
Koltzoff, N. K. 273, 462 
Koninckina 570 


‘Koodoo, horns of 624 


Koéppen, Wladimir 111 
Korotneff, A. 377 
Kraus;.G: 77 

Krogh, A. 109 

Krohl 666 

Kiihne, W. 235 
Kiister, E. 430 


Lafresnaye, F. de 653 
Lagena 251, 256, 260, 587 


Lagrange, J. L. 649 
Lalanne, L. 334 
Lamarck, J. B. de 
Lamb, A. B. 186 
Lamellaria 554 
Lamellibranchs 561 
Lami, B. 296, 643 
Laminaria 315 
Lammel, R. 100 
Lanchester, F. W. 26 
Lang, Arnold 561 
Lankester, Sir EH. 
465 
Laplace, P. 8S. de 1, 207,217 
Larmor, Sir J. 9, 259 
Lavater, J. C. 740 j 


5949, 716 


Ray 4, 251, 348, 


INDEX 


Law, Borelli’s 29; Brandt’s 482; of 
Constant Angle 599; Errera’s 213, 
306; Froude’s 22; Lamarle’s 
309: of Mass 130; Maupertuis’s 
208; Miiller’s 481; of Optimum 
110; van’t Hoff’s 109; Willard- 
Gibbs’ 280; Wolff’s 3, 51, 155 

Leaping 29 

Leaves, arrangement of 635; form of 
731 

Ledingham, J. C. G. 211 

Leduc, Stéphane 162, 167, 185, 219, 
259, 415, 428, 431, 590 

Leeuwenhoek, A. van 36 209 

Leger, L. 452 

Le Hello, P. 30 

Lehmann, O. 203, 272, 440, 485, 590 

Leibniz, G. W. von 3, 5, 159, 385 

Leidenfrost, J. G. 279 

Leidy, J. 252, 468 

Leiper, R. T. 660 

Leitch, I. 112 

Leitgeb, H. 305 

Length-weight coefficient 98-103, 775 

Leonardo da Vinci 27, 635; of Pisa 
643 

Lepeschkin 625 

Leptocephalus 87 

Leray, Ad. 18 

Lesage, G. L. 18 

Leslie, Sir John 

Lestiboudois, T. 

Leucocytes 211 

Levers, Orders of 690 

Levi, G. 35,°37 

Lewis, C. M: 280 

Lhuilier, S. A. J. 330 

Liesegang’s rings 427, 475 

Light, pressure of 48 

ealliesehe ue Ae 147. Sal: 
187, 192 

Lima 565 

Limacina 571 : 

Lines of force 163; of growth 562 

Lingula 251, 567 

Linnaeus 28, 250, 547, 720 

Lion, brain of 91 

Liquid veins 265 

Lister, Martin 318; 

Listing, J. B. 385 

Lithostrotion 325 

Littorina 524 

Lituites 546, 550 

Llama 703 

Lobsters’ claws 149 

Locke, John 6 

leper eh 1325 3b, 186,147, V5, 
191, 193 

Loewy, A. 281 

Logarithmic spiral, 493, ete. 


163, 503 
636 


R.S. 180, 


J. J. 436 


-1 
0 2) 
~] 


Loisel, G. 88 

Loligo, shell of 575 

Lo Monaco 83 

Loénnberg, E. 614, 632 
Looss, A. 660 

Lotze, R. H. 55 

Love, A. E. H. 674 

Lueas, F. A. 138 

Luciani, L. 83 

Lucretius 47, 71, 137, 160 
Ludwig, Carl 2; F. 643; H. J. 342 
Lupa 744: 

Lupinus, growth of 109, 112 


Macalister, A. 557 

MacAlister, Sir D. 673, 683 

Macallum, A. B. 277, 287, 357, 395; 
Je Ba 492 

McCoy, F. 388 

Mach, Ernst 209, 330 

Machaerodus, teeth of 633 

McKendrick, J. G. 42 

McKenzie, A. 418 

Mackinnon, D. L. 268 

Maclaurin, Colin 330, 779 

Macroscaphites 550 

Mactra 562 

Magnitude 16 

Maillard, L. 163 

Maize, growth of 109, 111, 298 

Mall, F. P. 492 

Maltaux, Mile 114 

Mammoth 634, 705 

Man, growth of 61; skull of 770 

Maraldi, J. P. 329, 473 

Marbled papers 736 

Marcus Aurelius 609 

Markhor, horns of 619 

Marsh, O. C. 706, 754 

Marsigli, Comte L. F. 

Massart, J. 114 

Mastodon 634 

Mathematics 719, 778 ete. 

Mathews, A. 285 

Matrix 656 

Matter and energy 11 

Matthew, W. D. 707 


de 652 


_Matuta 744 


Maupas, M. 133 

Maupertuis 3, 5, 208 

Maxwell, J. Clerk 9, 18, 40, 44, 160, 
207, 385, 691 

Mechanical efficiency 670 

Mechanism 5, 161, 185, ete 

Meek, C. F. U. 190 

Melanchthon 4 

Melanopsis 557 

Meldola, R. 670 

Melipona 332 


Mellor, J. W. 134 


788 INDEX 


Melo 525 

Melobesia 412 

Melsens, L. H. F. 282 

Membrane-formation 281 

Mensbrugghe, G. van der 212, 295, 470 

Meserer, O. 683 

Mesocarpus 289 

Mesohippus 766 

Metamorphosis 82 

Meves, F. 163, 285 

Meyer, Arthur 432; G. H. 8, 682, 683 

Micellae 157 

Michaelis, L. 277 

Miecrochemistry 288 

Micrococei 39, 245, 250 

Micromonas 38 

Miliolidae 595, 604 

Milner, R. S. 280 

Milton, John 779 

Mimicry 671 

Minchin, E. A. 267, 444, 449, 455 

Minimal areas 208, 215, 225, 293, 306, 

336, 349 

Minot, C. 8. 37, 72, 722 

Miohippus 767 

Mitchell, P. Chalmers 703 

Mitosis 170 

Mitra 557, 559 

Mobius, K. 449 

Modiola 562 

Mohl, H. von 624 

Molar and molecular forces 53 

Mole-cricket, chromosomes of 181 

Molecular asymmetry 416 

Molecules 41 

Moller, V. von 593 

Monnier, A. 78, 132 

Monticulipora 326 

Moore, B. 272 

Morey, 8. 264 

Morgan, T. H. 126, 134, 138, 147 

Morita 699 

Morphodynamique 156 

Morphologie synthétique 420 

Morphology 719, etc. 

Morse, Max 136 

Moseley, H. 8, 518, 521, 538, 553, 555, 
592 

Moss, embryo of 374; gemma of 403; 
rhizoids of 356 

Mouillard, L. P.. 27 

Mouse, growth of 82 

Mucor, sporangium: of 303 

Miillenhof, K. von 25, 332 

Miiller, Fritz 3; Johannes 459, 481 

Mummery, J. H. 425 

Munro, H. 323 

Musk-ox, horns of 615 

Mya 422, 561 

Myonemes 562 


Naber, H. A. 511, 650 

Nageli, C. 124, 159, 210 

Nassellaria 472 

Natica 554, 557, 559 

Natural selection 4, 58, 137, 456, 586, 
609, 651, 653 

Naumann, C. F. 529, 531, 539, 550, 
577, 594, 6363 "J. Fo6oa 

Nautilus 355, 494, 501, 515, 518, 532, 
535, 546, 552; 557, 575, 577, 580; 
592, 633; hood of 554; kidney 
of 425; N. umbilicatus 542, 547, 
554 

Nebenkern 285 

Neottia, pollen of 396 

Nereis, egg of 342, 378, 453 

Nerita 522, 555 

Neumayr, M. 608 

Neutral zone 674, 676, 686 

Newton 1, 6, 158, 643, 721 

Nicholson, H. A. 325, 327 

Noctiluca 246 

Nodoid 218, 223 

Nodosaria 262, 535, 604 

Norman, A. M. 465 

Norris, Richard 272 

Nostoe 300, 313 

Notosuchus 753 

Nuclear spindle 170; structure 166 

Nummulites 504, 552, 591 

Nussbaum, M. 198 


Oekotraustes 550 

Ogilvie-Gordon, M. M. 423 

Oil-globules, Plateau’s 219 

Oithona 742 

Oken, L. 4, 635 

Oliva 554 

Ootype 660 

Operculina 594 

Operculum of gastropods 521 

Oppel, A. 88 

Optimum temperature 110 

Orbitolites 605 

Orbulina 59, 225, 257, 587, 598, 604, 
607 

Organs, growth of 88 

Orthagoriscus 751, 775, 777 

Orthis 561, 567 

Orthoceras 515, 548, 551, 556, 579, 
735 

Orthogenesis 549 

Orthogonal trajectories 305, 377, 400, 
640, 678 

Orthostichies 649 

Orthotoluidene 219 

Oryx, horns of 616 

Osborn, H. F. 714, 727, 760 

Oscillatoria 300 

Osmosis 124, 287, etc. 


INDEX 


Osmunda 396, 406 

Ostrea 562 

Ostrich 25, 707, 708 

Ostwald, Wilhelm 44, 131, 426; 
Wolfgang 32, 77, 82, 132, 277, 281 

Otoliths 425, 432 

Ovis Ammon 614 

Owen, Sir R. 20, 575, 654, 669, 715 

Ox, cannon-bone of 730, 738; growth 
of 102 

Oxalate, calcium 412, 434 

Palaeechinus 663 

Palm 624 

Pander, C. H. 55 

Pangenesis 44, 157 

Papillon, Fernand 10 

Pappus of Alexandria 328 

Parabolic girder 693, 696 

Parahippus 767 

Paralomis 744 

Paraphyses of mosses 351 

Parastichies 640, 641 

Passiflora, pollen of 396 

Pasteur, L. 416 

Patella 561 

Pauli, W. 211, 484 

Pearl, Raymond 90, 97, 654 

Pearls 425, 431 

Pearson, Karl 36, 78 

Peas, growth of 112 

Pecten 562 

Peddie, W. 182, 272, 344, 448 

Pellia, spore of 302 

Pelseneer, P. 570 

Pendulum 30 

Peneroplis 606 

Percentage-curves, Minot’s 72 

Pericline 360 

Periploca, pollen of 396 

Peristome 239 

Permeability, magnetic 177, 182 

Perrin, J. 43, 46 

Peter, Karl 117 

Pettigrew, J. B. 490 

Pfeffer, W. 111, 273, 688 

Pfliiger, E. 680 

Phagocytosis 211 

Phascum 408 

Phase of curve 68, 81, ete. 

Phasianella 557, 559 

Phatnaspis 482 

Phillipsastraea 327 

Philolaus 779 

Pholas 561 

Phormosoma 664 

Phractaspis 484 

Phyllotaxis 635 

Phylogeny 196, 251, 548, 716 

Pike, #. H. 110 


789 


Pileopsis 555 

Pinacoceras 584 

Pithecanthropus 772 

Pith of rush 335 

Plaice 98, 105, 117, 432, 710, 774 

Planorbis 539, 547, 554, 557, 559 

Plateau, FE. 30,232; J. A. BF. 192; 212; 
218, 239, 275, 297, 374, 477 

Plato 2,478,720; Platonic bodies 478 

Plesiosaurs 755 

Pleurocarpus 289 

Pleuropus 573 

Pleurotomaria 557 

Plumulariidae 747 

Pluteus larva 392, 415 

Podocoryne 342 

Poincaré, H. 134 

Poiseuille, J. L. M. 669 

Polar bodies 179; furrow 

Polarised light 418 

Polarity, morphological 166, 168, 246, 
265, 284 

Pollen 396, 399 

Polyhalite 433 

Polyprion 749, 776 

Polyspermy 193 

Polytrichum 355 

Pomacanthus 749 

Popoff, M. 286 

Potamides 554 

Potassium, in living cells 288 

Potential energy 208, 294, 601, etc. 

Potter’s wheel 238 

Potts, R. 126 

Pouchet, G. 415 

Poulton, E. B. 670 

Poynting, J. H. 235 

Precocious segregation 

Preformation 54, 159 

Prenant, A. 163, 164, 189, 286, 289 

Prévost, Pierre 18 

Pringsheim, N. 377 

Probabilities, theory of 61 

Productus 567 

Protective colouration 671 

Protococcus 59, 300, 410 

Protoconch 531 

Protohippus 767 

Protoplasm, structure of 172 

Przibram, Hans 16, 82, 107, 149, 204, 
211, 418, 595; Karl 46 

Psammobia 564 

Pseuopriacanthus 749 

Pteranodon 756 

Pteris, antheridia of 409 

Pteropods of 258, 570 

Pulvinulina 514, 595, 600, 602 

Pupa 530, 549, 556 

Pitter, A. 110; 211, 492 

Pyrosoma, egg of 377 


310, 340 


348 


5U—3 


790 


Pythagoras 2, 509, 651, 720, 779 


Quadrant, bisection of 359 
Quekett, J. T. 423 

Quetelet, A. 61, 78, 93 

Quincke, G. H. 187, 191, 279, 421 


Rabbit, skull of 764. 

Rabl, K. 36, 310 

Radial co-ordinates 730 

Radiolaria 252, 264, 457, 467, 588, 607 

Rainey, George 7, 420, 431, 434 

Rainfall and growth 121 

Ram, horns of 613-624 

Ramsden, W. 282 

Ramulina 255 

Rankine, W. J. Macquorn 697, 712 

Ransom’s waves 164 

Raphides 412, 429, 434 

Raphidiophrys 460, 463 

Rasumowsky 683 

Rat, growth of 106 

Rath, O. vom 181 

Rauber, A. 200, 305, 310, 380, 382, 398, 
677, 683 

Ray, John 3 

Rayleigh, Lord 438, 44 

Réaumur, R. A. de 8, 108, 329 

Reciprocal diagrams 697 

Rees, R. van 374 

Regeneration 138 

Reid, E. Waymouth 2 

Reinecke, J. C. M. 528 

Reinke, J. 303, 305, 355, 356 

Reniform shape 735 

Reticularia 569 

Reticulated patterns 258 

Réticulum plasmatique 468 

Rhabdammina 589 

Rheophax 263 

Rhinoceros 612, 760 

Rhumbler, L. 162, 165, 260, 322, 344, 
465, 466, 589, 590, 595, 599, 608, 
628 

Rhynchonella 561 

Riccia, 372, 403, 405 

Rice, J. 242, 273 

Richardson, G. M. 416 

Riefstahl, E. 578 

Riemann, B. 385 

Ripples 33, 261, 323 

Rivularia 300 

Roaf, Hs C57 272 

Robert, A. 306, 339, 348, 377 

Roberts, C. 61 

Robertson, T. B. 82, 132, 191, 192 

Robinson, A. 681 

Roérig, A. 628 

Rose, Gustay 421 

Rossbach, M. J. 165 


INDEX 


Rotalia 214, 535, 602 

Rotifera, cells of 38 

Roulettes 218 

Roux, W. 8, 55, 57, 157, 194, 378, 
383, 666, 683 

Ruled surfaces 230, 270, 582 

Ruskin, John 20 

Russow, — 73, 75 

Ryder, J. A. 376 


Sachs, J. 35, 38, 95, 108, 110, 111, 
200, 360, 398, 399, 624, 635, 640, 
651, 680 

Sachs’s rule 297, 300, 305, 347, 376 

Saddles, of ammonites 583 

Sagrina 263 

St Venant, Barré de 621, 627 

Salamander, sperm-cells of 179 

Salpingoeca 248 

Salt, crystals of 429 

Salvinia 377 

Samec, M. 434 

Samter, M. and Heymons 130 

Sandberger, G. 539 

Sapphirina 742 

Saville Kent, W. 246, 247, 248 

Scalaria 526, 547, 554, 557, 559 

Seale, effect of 17, 438 

Scaphites 550 

Scapula, human 769 

Scarus 749 

Schacko, G. 604 

Schaper, A. A. 83 

Schaudinn, F. 46, 286 

Scheerenumkehr 149 

Schewiakoff, W. 189, 462 

Schimper, C. F. 502, 636 

Schmaltz, A. 675 

Schmankewitsch, W. 130 

Schmidt, Johann 85, 87, 118 

Schonflies, A. 202 

Schultze, F. E. 452, 454 

Schwalbe, G. 666 

Schwann, Theodor 

Schwartz, Fr. 172 

Schwendener, 8. 210, 305, 636, 678 

Scorpaena 749 

Scorpioid cyme 502 

Scott, E. L. 110; W. B. 768 

Scyromathia 744 

Searle, H. 491 

Sea urchins 661; egg of 173; growth 
of 117, 147 

Sebastes 749 

Sectio aurea 511, 643, 649 

Sedgwick, A. 197, 199 

Sédillot, Charles E. 688 

Segmentation of egg 57, 310, 344, 382, 
etc.; spiral do., 371, 453 

Segner, J. A. von 205 


199, 380, 591 


INDEX 


Selaginella 404 
Semi-permeable membranes 272 
Sepia 575, 577 

Septa 577, 592 

Serpula 603 

Sexual characters 135 

Sharpe, D. 728 

Shearing stress 684, 730, etc. 
Sheep 613, 730, 738 

Shell, formation of 422 
Sigaretus 554 

Silkworm, growth of 83 
Similitude, principle of 17 
Sims Woodhead, G. 414, 434 
Siphonogorgia 413 

Skeleton 19, 438, 675, 691, etc. 
Snow crystals 250, 480, 611 
Soap-bubbles 43, 219, 299, 307, etc. 
Socrates 8 


Sohncke, L. A. 202 
Solanum 625 

Solarium 547, 554, 557, 559 
Solecurtus 564 

Solen 565 


Sollas, W. J. 440, 450, 455 

Solubility of salts 434 

Sorby, H. C. 412, 414, 728 

Spallanzani, L. 138 

Span of arms 63, 93 

Spangenberg, Fr. 342 

Specific characters 246, 380; 
tive capacity 177; 
215 

Spencer, Herbert 18, 22 

Spermatozoon, path of 193 

Sperm-cells of Crustacea 273 

Sphacelaria 351 

Sphaerechinus 117, 147 

Sphagnum 402, 407 

Sphere 218, 225 

Spherocrystals 434 

Spherulites 422 

Spicules 282, 411, etc. 

Spider’s web 231 

Spindle, nuclear 169, 174 

Spinning of protoplasm 164 

Spiral, geodetic 488; logarithmic 493, 
etc.; segmentation 371, 453 

Spireme 173, 180 

Spirifer 561, 568 

Spirillum 46, 253 

Spirochaetes 46. 230, 266 

Spirographis 586 

Spirogyra 12, 221, 227, 242, 244, 275, 
287, 289 

Spirorbis 586, 603 

Spirula 528, 547, 554, 575, 577 

Spitzka, E. A. 92 

Splashes 235, 236, 254, 260 

Sponge-spicules 436, 440 


induc- 
surface 32, 


791 


Spontaneous generation 420 
Sporangium 406 
Spottiswoode, W. 779 
Spray 236 

Stallo, J. B. 1 

Standard deviation 78 
Starch 432 

Starling, E. H. 135 
Stassfurt salt 433 
Stegocephalus 746 


Stegosaurus 706, 707, 710, 754 


Steiner, Jacob 654 

Steinmann, G. 431 

Stellate cells 335 

Stentor 147 

Stereometry 417 

Sternoptyx 748 

Stillmann, J. D. B. 695 

St Loup, R. 82 - 

Stokes, Sir G. G. 44 

Stole, Ant. 452 

Stomach, muscles of 490 

Stomata 393 

Stomatella 554 

Strasbiirger, E. 35, 283, 409 

Straus-Diirekheim, H. E. 30 

Stream-lines 250, 673, 736 

Strength of materials 676, 679 

Streptoplasma 391 

Strophomena 567 

Studer, T. 413 

Stylonichia 133 

Succinea 556 

Sunflower 494, 635, 639, 688 

Surface energy 32, 34, 191, 207, 278, 
293, 460, 599 

Survival of species 251 

Sutures of cephalopods 583 

Swammerdam, J. 8, 87, 380, 528, 585 

Swezy, Olive 268 

Sylvester, J. J. 723 

Symmetry, meaning of 209 

Synapta, egg of 453 

Syncytium 200 

Synhelia 327 

Szielasko, A. 654 


Tadpole, growth of 83, 114, 138, 153 
Tait, P. G. 35, 43, 207, 644 

Taonia 355, 356 
Tapetum 407 
Tapir 741, 763 
Taylor, W. W. 277, 
Teeth 424, 612, 632 
Telescopium 557 
Telesius, Bernardinus 656 
Tellina 562 

Temperature coefficient 109 
Terebra 529, 557, 559 
Terebratula 568, 574, 576 


282, 426, 428 


792 


Teredo 414 

ernie 

Terquem, O. 329 

Tesch, J. J.- 573 

Tetractinellida 443, 450 

Tetrahedral symmetry 315, 396, 476 

Tetrakaidecahedron 337 

Tetraspores 396 

Textularia 604 

Thamnastraea 327 

Thayer, J. EH. -672 

Thecidium 570 

Thecosmilia 325 

Theel, H. 451 

Thienemann, F. A. L. 653 

Thistle, capitulum of 639 

Thoma, R. 666 

Thomson, James 18, 259; J. A. 465; 
J. J. 235, 280; Wryville 466 

Thurammina . 256 

Thyroid gland 136 

Time-element 51, 496, 
energy diagram 63 

Tintinnus 248 

Tissues, forms of 293 

Titanotherium 704, 762 

Tomistoma 753 

Tomlinson, C. 259, 428 

Tornier, G. 707 

Torsion 621, 624 

Trachelophyllum 249 

Transformations, theory of 562, 719 

Traube, M. 287 

Trees, growth of 119; height of 19 

Trembley, Abraham 138, 146 

Treutlein, P. 510 

Trianea, hairs of 234 

Triangle, properties of 508; of forces 
295 

Triasters 327 

Trichodina 25: 

Trichomastix 

Triepel, H. 683, 684 

Triloculina 595 

Triton 554 

Trochus 377, 557, 560; 
of 340 

Trondle, A. 625 

Trophon 526 

Trout, growth of 94 

Trypanosomes 245, 266, 269 

Tubularia 125, 126, 146 

Turbinate shells 534 

Turbo 518, 555 

Turgor 125 

Turner, Sir W. 769 

Turritella 489, 524, 
559 

Tusks 515, 612 

Tutton, A. E. H. 202 


etc.; time- 


embryology 


INDEX 


Twining plants 624 
Tyndall, John 428 


Umbilicus of shell 547 
Underfeeding, effect of 106 
Undulatory membrane 266 
Unduloid 218, 222, 229, 246, 256 
Unio 341 

Univalve shells 553 

Urechinus 664 


Vaginicola 248 
Vallisneri, Ant. 138 
Van Iterson, G. 595 
Van Rees, R. 374 
Van't Hof, J:°H: 
Variability 78, 103 
Venation of wings 385 
Verhaeren, Emile 778  . 
Verworn, M. 198, 211, 467, 605 
Vesque, J. 412 

Vierordt, K. 73 

Villi 32 

Vincent, J. H. 323 

Vines, S. H. 502 

Virchow, R. 200, 286 

Vital phenomena 14, 417, ete. 
Vitruvius 740 

Volkmann, A. W. 669 

Voltaire 4, 146 

Vorticella 237, 246, 291 


1, 110, 433 


Wager, H. W. T. 259 
Walking 30 
Wallace, A. R. 5, 
Wallich-Martius 77 
Warburg, O. 161 
Warburton, C. 233 
Ward, H. Marshall 133 
Warnecke, P. 93 
Watase, 8. 378 
Water, in growth 125 
Watson, F. R. . 323 : 
Weber, E. H. 210, 259, 669; 
and W. E. 30; Max 91 
Weight, curve of 64, ete. 
Weismann, A. 158 
Werner, A. G. 19 
Wettstein, R. von 728 
Whale, affinities 716; size 21; struc- 
ture 708 
Whipple, I. L. 123 
Whitman, C. O. 157, 164, 198, 194, 
199, 200 


432, 549 


Hee 


Whitworth, W. A. 506, 512 
Wiener, A. F. 45 
Wildeman, E. de 307, 355 : 


Willey, A. 425, 548, 555, 578 
Williamson, W. C. 423, 609 
Willughby, Fr. 318 


INDEX 


Wilson, E. B. 150, 163, 173, 195, 199, 
311, 341, 342, 398, 453 

Winge, .O. 433 

Winter eggs 283 

Wissler, Clark 79 

Wissner, J. 636 

Wohler, Fr. 416, 420 

Wolff, J. 683; J.C. F. 3, 51, 155 

Wood, R. W. 590 

Woods, R. H. 666 

Woodward, H. 578; S. P. 554, 567 

Worthington, A. M. 235, 254 

Wreszneowski, A. 249 

Wright, Chauncey 335 

Wright, T. Strethill 219 

Wyman, Jeffrey 335 


793 


Yeast cell 213, 242 

Yield-point 679 

Yolk of egg 165, 660 

Young, Thomas 9, 36, 669, 691 


Zangger, H. 282 
Zeising, A. 636, 650 
Zeleny, C. 149 
Zeuglodon 716 
Zeuthen, H. G. 511 
Ziehen, Ch. 92 
Zittel, K. A. von 325, 327, 548, 584 
Zoogloea 282 
Zschokke, F. 683 
Zsigmondy 39 
Zuelzer, M. 165 


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